Josephson effects in carbon nanotube mechanical resonators and

Josephson e�ects in carbon nanotube mechanical resonators and graphene Han Keijzers

Josephson e�ects in carbon nanotube mechanical resonators and graphene

Han Keijzers

Uitnodiging

Josephson e�ects in carbon nanotubemechanical resonators

and graphene

voor het bijwonen van de openbareverdediging van mijn proefschrift

en bijbehorende stellingen

op maandag 15 oktoberom 15:00 uur in de Aula

van de TU DelftMekelweg 5 te Delft

Een half uur voor aanvang (14:30 uur) zal ik het onderwerp van

mijn promotie kort toelichten.

Han KeijzersBoerhaavelaan 114

2334 ET Leiden

Paranimfen:Vincent Mourik, 06 3061 0308

Ciprian Padurariu, 06 4511 8617


and graphene

Han Keijzers

A carbon nanotube is a unique one dimensional tube in which the quantum nature of electrons can be studied. When cooleddown to temperatures close to absolute zero, a nanotube can become superconducting by connecting it to a superconductor. Due to the Josephson e�ect, a dissipationless supercurrent oscillating at several GHz, can �ow 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 e�ect of magnetic �eld on supercurrent in graphene, and present our experimental results.

Casimir PhD Series, Delft-Leiden 2012-19ISBN: 978-90-8593-130-0



Han Keijzers

Uitnodiging


and graphene








2334 ET Leiden




and graphene

Han Keijzers







Han Keijzers

Uitnodiging


and graphene








2334 ET Leiden




and graphene

Han Keijzers







Han Keijzers

Uitnodiging


and graphene








2334 ET Leiden




and graphene

Han Keijzers





Propositions

belonging to the thesis

Josephson effects in carbon nanotube mechanical resonators andgraphene

C.J.H. Keijzers

1. The presence of the AC Josephson effect enhances transduction of mechan-ical displacement to electrical signals by up to two orders of magnitude.Chapter 5 of this thesis.

2. Even during a complete suppression of the observable DC supercurrent, theamplitude and sign of AC Josephson currents can still be detected.Chapter 5 of this thesis.

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 theresonator in comparison to the large linewidth of the Josephson current.Chapter 5 of this thesis.

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 legitimacyof all arguments of prosecutor and defender must be subject to review bylawyers 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 schooledpersonnel 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 “reconstructionmission” by government and press increases misunderstanding of psycho-logical problems in veterans.

9. A speaker who adds a progress indicator to his PowerPoint presentationwill never make his audience so engrossed in his subject that they lose thesensation 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.

Delft, September 2012

Stellingen

behorende bij het proefschrift

Josephson effecten in koolstofnanobuis mechanische resonatorenen grafeen

C.J.H. Keijzers

1. De aanwezigheid van het AC Josephson effect verhoogt de transductie vanmechanische verplaatsing naar elektrische signalen met tot wel twee ordesvan 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 steedsgedetecteerd worden.Hoofdstuk 5 van dit proefschrift.

3. De voornaamste reden voor de inefficiente energie uitwisseling tussen eenkoolstofnanobuis resonator en de AC Josephson stroom is de kleinelijnbreedte van de resonator ten opzichte van de grote lijnbreedte van deJosephson stroom.Hoofdstuk 5 van dit proefschrift.

4. Het verbieden van de vereniging Martijn∗ is in strijd met de beschermingvan de openbare orde.∗Uitspraak Rechtbank Assen, 27 juni 2012.

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

6. Creatieve technische oplossingen kunnen alleen ontstaan in complete artistiekevrijheid, terwijl innovatie alleen mogelijk is door de zorgvuldige begrenzingdaarvan.

7. De beste manier om het dreigend tekort∗ aan technisch geschoold personeelte 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 problemenbij 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.

Delft, september 2012

Josephson effects in carbon nanotube

mechanical resonators

and graphene

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.

Dit proefschrift is goedgekeurd door de promotor:

Prof. dr. ir. L. P. Kouwenhoven

Samenstelling promotiecommissie:

Rector Magnificus, voorzitterProf. dr. ir. L.P. Kouwenhoven, Technische Universiteit Delft, promotorProf. dr. ir. J.E. Mooij, Technische Universiteit DelftProf. dr. Y.V. Nazarov, Technische Universiteit DelftProf. dr. ir. L.M.K. Vandersypen, Technische Universiteit DelftProf. dr. J. Aarts, Universiteit LeidenProf. dr. ir. H. Hilgenkamp, Universiteit TwenteDr. ir. G.A. Steele, Technische Universiteit DelftProf. 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. KeijzersCover design: C.J.H. KeijzersFront: 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

Copyright 2012 by C.J.H. KeijzersCasimir PhD Series, Delft-Leiden 2012-19ISBN: 978-90-8593-130-0An electronic version of this thesis is available at www.library.tudelft.nl/dissertations

Contents

1 Introduction 11.1 Quantum nanoscience . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Carbon based nano-electronics . . . . . . . . . . . . . . . . . . . . 21.4 Nanomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Clean carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 8Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Theoretical concepts 152.1 Carbon nanotube quantum dots . . . . . . . . . . . . . . . . . . . . 152.2 Andreev reflection and supercurrent . . . . . . . . . . . . . . . . . 272.3 Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Carbon nanotube mechanical resonator . . . . . . . . . . . . . . . 402.5 Vibrating suspended carbon nanotube Josephson junctions . . . . 442.6 Josephson junctions in a magnetic field . . . . . . . . . . . . . . . . 51Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Device fabrication 653.1 Electron beam lithography . . . . . . . . . . . . . . . . . . . . . . . 653.2 Fabrication of graphene Josephson junctions . . . . . . . . . . . . . 653.3 Fabrication of suspended carbon nanotube Josephson junctions . . 683.4 Selecting good nanotube devices . . . . . . . . . . . . . . . . . . . 74Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Graphene Josephson junctions 814.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Proposed experiments on Zeeman π-junctions in graphene . . . . . 824.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 854.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

v

Contents

5 Vibrating suspended clean carbon nanotube Josephson junctions 955.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.3 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.4 Shapiro steps at the mechanical resonance frequency . . . . . . . . 1055.5 Mixing or rectification? . . . . . . . . . . . . . . . . . . . . . . . . 1115.6 Signal power dependence . . . . . . . . . . . . . . . . . . . . . . . . 1155.7 Magnetic field dependence . . . . . . . . . . . . . . . . . . . . . . . 1205.8 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . 1345.9 Mechanical resonance at Shapiro plateaus . . . . . . . . . . . . . . 1375.10 Observed features and general conclusions . . . . . . . . . . . . . . 141Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6 Mechanical resonance at a fractional driving frequency 1476.1 Parametric excitation and detection by Josephson mixing . . . . . 1476.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7 Future directions for superconducting CNT resonators 1637.1 Current status and main challenges . . . . . . . . . . . . . . . . . . 1637.2 Coupling CNT motion to a transmon qubit . . . . . . . . . . . . . 1657.3 Coupling CNT motion to superconducting LC resonators . . . . . 1707.4 Josephson parametric amplifier with a suspended CNT junction . . 1747.5 High magnetic field compatible CNT Josephson junctions . . . . . 1787.6 Consideration of future directions . . . . . . . . . . . . . . . . . . . 180Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

A Additional data 187A.1 Additional data on graphene . . . . . . . . . . . . . . . . . . . . . 187A.2 Additional data on carbon nanotubes . . . . . . . . . . . . . . . . 188Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

B Vibrating carbon nanotube quantum dots 201B.1 Effects on conductance by displacement . . . . . . . . . . . . . . . 201B.2 Conductance through a QD in the presence of oscillating voltages . 207Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

C The effect of mechanical resonance on Josephson dynamics 213C.I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215C.II The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217C.III Coupling and non-linearities . . . . . . . . . . . . . . . . . . . . 219C.IV Phase bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223C.V D.C. voltage bias . . . . . . . . . . . . . . . . . . . . . . . . . . . 225C.VI Shapiro steps at resonant driving . . . . . . . . . . . . . . . . . . 227C.VII Shapiro steps at non-resonant driving . . . . . . . . . . . . . . . 230C.VIII Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232C.A Lagrangian formalism . . . . . . . . . . . . . . . . . . . . . . . . 233Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

vi

Contents

D Characterization of rhenium films 237D.1 Experimental goal . . . . . . . . . . . . . . . . . . . . . . . . . . . 237D.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237D.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239D.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

E Fabrication recipes 243E.1 Fabrication of graphene Josephson junctions . . . . . . . . . . . . . 243E.2 Fabrication of suspended carbon nanotube Josephson junctions . . 246

F Superconducting magnet coil 257

Summary 259

Samenvatting 261

Acknowledgements 263

Curriculum Vitae 267

vii

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 areperformed in conditions that permit observation of the quantum nature of theelectrons in the system, or of the structure itself.

The development of nanotechnology in the last two decades makes it now possibleto build structures where quantum mechanical effects become important. It offersa toolbox with which it is possible to make and measure structures in which,for example, the wave nature of electrons dominates the conductance. Typicaldimensions 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 100mK.

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 ofcomputer that uses quantum mechanics to perform calculations in a completelydifferent way compared to ordinary computers. When a QC is sufficiently large,it can outperform ordinary computers on specific tasks. Information in quantumcomputers is stored in quantum bits (or qubits) rather than conventional bits.Small quantum computers are already around in research labs, and much effortis made to investigate existing building blocks, develop new types, and combinethem to make larger quantum computers. A review on superconducting qubitscan be found in Ref. [1].

On the road towards the quantum computer, many interesting things can belearned about the nature of electrons and the devices in which they reside. Wehave 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 donein new regimes where these materials have not been operated before. The tech-niques we have developed, together with our experimental observations, can beuseful in future quantum electronic devices.

1

1. Introduction

In this thesis we will present the results of two experiments. The first experimentis on graphene Josephson junctions in large magnetic fields. Our goal is to realizea π-junction in graphene, by means of a magnetic field. Such junctions havespecific 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 broughtclosely together in a single suspended carbon nanotube (CNT). These fields arenanomechanics 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 practical applications of such resonators, and they arealso 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 losetheir resistance. In the superconducting state, a superconductor can carry a cur-rent without dissipating energy. This is called a supercurrent. Aluminum andniobium are two common superconductors. In these (and many other) metalsthe interaction of electrons with phonons causes pairing of electrons in Cooperpairs, that form a condensate that can carry a supercurrent. This macroscopicquantum-mechanical effect occurs when the thermal energy of the system is belowthe binding energy associated to Cooper pairing [2].

To observe the superconducting state a superconductor has to be cooled belowits critical temperature. Superconductivity can then be inferred by measuringthe electrical resistance. This was first done in Leiden, by Heike KamerlinghOnnes, who discovered superconductivity in mercury in 1911 [3]. The discoveryof 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 methoda (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 toaccess, but also to define the structure. The advantage of the bottom-up methodis 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

2

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 carbonatoms was extracted by mechanical exfoliation from graphite [8, 9]. In Fig. 1.1we give an overview of carbon allotropes. Carbon nanotubes and graphene haveexceptional electronic and mechanical properties. This has spurred a great effortby 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 leftto 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 bandgapdepending on its chirality (the way graphene is rolled up to form a nanotube).Small bandgap nanotubes are called metallic nanotubes and have gaps on theorder of 50meV. Large bandgap nanotubes have gaps on the order of 500meV.

3

1. Introduction

A nanotube based FET was first demonstrated in 1998 [12]. A lot of interest isin the development of CNT FETs, because of their exceptional properties.

Carbon nanotubes can have a very high mobility, exceeding 105 cm2/Vs at roomtemperature. This is much higher than the mobility of graphite (∼ 2·104 cm2/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 itharder for them to scatter [13]. Because in addition CNTs have very few struc-tural defects, they behave as ballistic conductors. For these reasons, CNTs makeexceptionally good FETs. Recently a sub-10 nm transistor made from a CNTwas reported that is smaller and performs better than current silicon transistortechnology [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 CNTquantum dots can be used as building blocks for a quantum computer. For thisreason, 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 canflow 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 adevice is called a Josephson junction. The most common junction is made byemploying a natural oxide on a superconductor as a weak link, in a sandwichstructure (for example Al/Al2O3/Al). Nanotubes can also be used as weak link,and have certain advantages over typical Josephson junctions. We point out twoadvantages of CNT junctions: Gate tunable supercurrents and small dimensions.

Because the CNT is a semiconductor, its conductance can be changed with agate. In this way the coupling between the two superconductors (and with thisthe maximum supercurrent) can be changed in situ, which is not possible withan oxide junction. The first superconducting CNT transistors were made in2006 [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 broughtvery close to magnetic molecules. This allows in principle for extreme sensitiv-ity to the magnetic flux generated by such molecules. This sensitivity could beemployed to characterize molecules, which is one application of such a device [20].

Josephson junctions have strong non-linear IV characteristics. The current througha Josephson junction is determined by the phase difference between the contactsrather than the voltage bias. As will be discussed in Ch. 2, the phase is also afunction 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

4

1.3. Carbon based nano-electronics

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

1.3.5 Special properties of graphene

Since its discovery, the unique properties of graphene result in a continuousstream 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 Schrodingerequation [22]. As a result, electrons behave differently in graphene compared toany other solid state system. Examples of this are the half-integer quantum Halleffect and Klein tunneling through a high potential barrier with approaching100% transmission probability for certain angles [9, 23–25].

Graphene is a semi-metal (zero-gap semiconductor). Its bandstructure looks likea Dirac cone. Because of the absence of a bandgap it is not possible to createquantum dots in as-is graphene. However, a bandgap can be introduced in bilayergraphene 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 beenachieved by a few groups now [29–31]. Many efforts have been made to reducethe disorder in graphene and improve its mobility. Several approaches have beenfollowed 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 105 cm2/Vs(at reduced temperature) can be achieved. Large sheets of graphene (∼ 80 cm)can be grown in an industrial setting, and (industrial scale) graphene transistorsare 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 timereversal symmetry (TRS) were absent or strongly suppressed in some graphenesamples [36]. Since both phase coherence and TRS are necessary for Andreevreflection, it was not expected that graphene could support supercurrents. Fur-thermore, since intrinsic graphene has a vanishing density of states at the Diracpoint, carrying supercurrent there was considered to be unfavorable also becauseof this reason.

5

1. Introduction

This was disproved in an experiment where graphene was contacted to two super-conducting leads, forming a weak link Josephson junction [37]. The experimentshowed that the maximum supercurrent can be gate tuned, and carried by holesas well as electrons. Transport in graphene is phase coherent even at the Diracpoint, where the supercurrent was still finite. Graphene Josephson junctions areunique, because they are the only junctions where the weak link is truly 2D. Thisproperty 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 thedomain of single electrons and atoms. The advance of nanotechnology has madeit possible to test on-chip the limits of quantum mechanics for systems witha large number of atoms. One way of distinguishing quantum behavior fromclassical 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 itcould be a superposition of position. In the macroscopic, classical world of tablesand chairs such exotic phenomena are not observed. Quantum nanomechanicsoffers a platform to study the emergence of the classical world from quantummechanics [38].

Since 2010 the first experiments have been reported where a mechanical resonatorhas 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 tosystems 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 rateof macroscopic superposition states, we can learn about the fundamental originof decoherence [43]. Experiments are proposed in which this is used to directlytest the limits of quantum theory [44].

A mechanical resonator becomes a quantum resonator when its average thermaloccupation (n = [exp(�ωr/kBT )− 1]−1) drops below one. In practice, this meansthat 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 > 1GHz) resonator in a dilution fridge (T < 50mK). However, such highfrequency resonators will have small zero point motion and hence their motionwill be hard to detect. As will be discussed in the next subsection, CNTs arevery promising nanomechanical systems (NEMS), because they combine highfrequency modes with large zero point motion and low damping. Much effortis been made now to make the first quantum resonator in a suspended carbonnanotube.

6

1.5. Clean carbon nanotubes

1.4.2 Carbon nanotube mechanics

From an engineering viewpoint, CNT NEMS are interesting systems because theyare the most sensitive force and mass sensors. Mass detection with a resolutionon the order of 1.7 yoctogram (1 yg = 10−24 g) has recently been achieved usinga 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 = 1.25TPa)and low mass density (ρ = 1350 kg/m3) of a CNT [46, 47]. In general, a large Eimplies a large spring constant and a high stiffness. In combination with a smallmass (on the order of m = 2 × 10−18 g = 2 ag), this results in high resonancefrequencies. In CNTs bending modes have now been detected with frequenciesin the 100MHz to 40GHz range [48–51]. In this thesis we will report on CNTswith fr ∼ 1GHz.

Ultimately the force sensitivity of a mechanical resonator is limited by its zeropoint motion (zpm). From the engineering viewpoint of building a quantumresonator 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 combinea (relatively) large zpm (∼ 1 pm) with large eigenfrequencies (∼ 1GHz) and forthis reason they are very suitable for embedding a quantum resonator [38].

An additional advantage of bottom-up CNT resonators compared to top-downfabricated NEMS, is their lack of structural and surface defects. It is likely thatthe origin of damping of mechanical vibrations lies in such defects in top-downfabricated NEMS [52]. Small damping implies long ring down times. In a quan-tum nanotube resonator, this could be used to manipulate and store quantumstates.

1.5 Clean carbon nanotubes

The outstanding intrinsic electronic and mechanical properties of CNT deviceswere 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 dirtyduring fabrication. The development of a new fabrication method, where theCNT 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 andmechanical properties. Quantum dots made in clean nanotubes can very easilybe tuned in the few electron/hole regime and are very stable. The mechanicalproperties are also outstanding, because they combine fundamental resonancefrequencies on the order of f ∼ 1GHz with very large quality factors, exceed-ing Q = 105. This discovery boosted the potential of CNTs in the field ofnanomechanics, because such high Q · f product has not been achieved in othernanomechanical systems [51].

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

7

1. Introduction

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

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

In the new type of clean fabrication, the nanotube is grown on a mother chipand then stamped onto a receiver chip on which all the gates and contacts havebeen pre-fabricated [56,57]. This new type of fabrication offers advantages abovethe existing one. The first advantage is that the structures on the receiver chipdo not have to be compatible with the CNT growth. The second advantage isthat a single CNT can (in principle, this has not been done yet because it is hardto see exactly where the tube is on the mother chip) be picked and placed on apredefined position. The conventional clean fabrication method which we haveused 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 notdiscuss many theoretical concepts in the experimental chapters, but refer to thetheory chapter when necessary. Chapter 3 is on the nanofabrication of grapheneand 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) experimenton suspended nanotube Josephson junctions. In the experiment on graphene,the dependence of supercurrent on an in-plane magnetic field is studied with thegoal of creating a Zeeman π-junction. This work is presented in Ch. 4. In Ch. 5we report on our study of suspended CNT Josephson junctions with GHz res-onance frequencies. We report on a novel mixing signal that we contribute toAC Josephson dynamics. In Ch. 6 we discuss an experiment in which we probethe superconducting CNT mechanical vibrations by driving the system at a sub-harmonic of the mechanical mode. In Ch. 7 we discuss the current status andchallenges regarding CNT resonators, and suggest a broad range of experimentswith suspended CNT Josephson junctions. I conclude Ch. 7 with a personalconsideration of future directions.

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

8

1.6. Outline of this thesis

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

We present an analysis of resonant interaction between AC Josephson effect andCNT 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 ourpaper we focus on a regime of resonant interaction in which energy is efficientlytransferred between resonator and supercurrent. Our experiment is probably notin this regime, due to a large mismatch between the linewidth of AC Josephsondynamics and the linewidth of the resonator. This will be discussed more indetail at the end of Sec. 5.10.

9

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13

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 resonators (Sec. 2.4).In a separate section we will consider interaction between Josephson effect andnanomechanics (Sec. 2.5). In the final section of this chapter (Sec. 2.6) we discussthe magnetic field dependence of the Josephson current. Properties of graphenewill 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 holesdominates their behavior. Nanotubes are narrow hollow cylinders made entirelyout of carbon, in which electrons or holes are naturally confined. In this sectionwe will discuss the nature of electrons and holes in graphene and CNTs, anddiscuss 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 hexagonallattice that gives graphene and CNTs a unique electronic structure: An electron-hole symmetric linear dispersion relation near two points, that are referred to asK and K′. We will see that this has important consequences for the behavior ofelectrons 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 fieldand spin-orbit interaction, and we will conclude with a short discussion on thebandgap of CNTs and the effect of longitudinal quantization.

15

2. Theoretical concepts

Graphene bandstructureA CNT can be modeled as a rolled up sheet of graphene. The bandstructureof CNTs is very similar to that of graphene, but is modified by quantizationconditions that are due to the spatial confinement of electrons. To understandthe bandstructure of CNTs it is necessary to first understand the bandstructureof 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 Aand 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 itshares one electron forming sp2 σ-bonds. The fourth valence electron occupies apz orbital. The allowed range of energies of this electron defines the bandstruc-ture. All pz orbitals can mix together, forming delocalized electron states thatdetermine the conductivity. The bandstructure can be found with a tight-bindingapproximation, in which only the nearest neighbor coupling of pz orbitals has tobe considered [1]. Instead of reproducing the mathematical derivation we willgive the outcome and comment on some important consequences. We will usea graphical approach to discuss how the CNT bandstructure appears from thegraphene bandstructure.

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

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

The pseudospin describes the amplitude of the electronic wavefunction on thesublattice atoms A and B. It can be shown that a pseudospin can be defined insuch a way that it points parallel to the direction of propagation k for electronsnear the K point, and antiparallel near the K′ point [4]. The pseudospin giveselectrons in graphene a chirality, or “handedness”. States close to K correspondto right-handed electrons with pseudospin parallel to k and states close to K′

are left-handed, with pseudospin antiparallel to k. The pseudospin assignmentis reversed for holes. One important implication of pseudospin is the suppressionof backscattering. In 1D metallic CNTs backscattering corresponds to scatteringfrom kx to −kx. It can be shown that this is forbidden, because the overlapintegral of two antiparallel pseudospins in the same valley is zero [4]. This is onereason for ballistic transport in metallic CNTs.

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

16

2.1. Carbon nanotube quantum dots

E

-EFkx ky

E

k||

k

valence

conductionB

A

x

ya

kx

ky

K’ K

K’ K

Δk=μμμμB/ ννννFBII

E

cw

ccw

ccw

A B C

D E F

k

kII

k

BII=0BII>0

k

μ μ μ μ II BcwEG

Figure 2.1: Electronic band structure of carbon nanotubes. (a) The unitcell 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 K′ pointswhere the dispersion can be approximated by Dirac cones. (c) Wavevectors of electronstraveling around the CNT (k⊥) have periodic boundary conditions which results inquantization. This set of discrete states is depicted with red lines. Each line is a 1Dsubband. (d) In a small bandgap nanotube, the quantization lines almost pass througha K point. The offset determines the bandgap Eg at B = 0T. (e) When a magneticfield is applied parallel to the CNT, an AB phase modifies the quantization conditionsby a shift Δk. This decreases the bandgap associated with K electrons and increases thebandgap for K′ electrons. (f) When contributions due to Zeeman splitting are ignored,each electronic state shifts in energy according to its orbital magnetic moment. Forexample, 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 captionare adapted from Ref. [2].

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

The states of electrons close to the charge neutrality points and with wavenumberk = K+ κ are described by the Dirac Hamiltonian:

H = �vFσ · κ , (2.1)

here vF ≈ 106 m/s is the Fermi velocity and σ are the Pauli matrices acting onthe pseudospin. The eigenvalues of the Dirac Hamiltonian are E = ±�vF|κ|.The eigenvectors of the Dirac Hamiltonian are the pseudospinors. At the chargeneutrality points graphene has a linear dispersion relation, which means that theeffective mass of electrons/holes is zero. A consequence of this is that electronsin graphene always move at the same velocity. In this sense electrons in graphene

17


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

Electron-wave quantization in a CNTDue to its cylindrical shape electron waves propagate in spirals on the surfaceof a CNT. This results in two quantization conditions. The first quantizationcondition is due to the phase accumulation along its circumference. This leads toa quantization of k⊥, the wave number perpendicular to the CNT axial direction.The second quantization condition is due to the phase accumulation along thelongitudinal direction and leads to a quantization of the wavenumber k‖. Thequantized 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 = B‖πD2/4 is the magnetic flux threading the CNT. The first term in thisequation is the dynamical phase that the electron wave picks up by traveling adistance πD (moving around the CNT once). The second term is the Aharonov-Bohm (AB) phase that the electron wave picks up by encircling the magneticflux [5].

Because the diameter of a CNT is very small (∼ 1 nm) compared to its length(∼ 1μm), electrons are free to move over much larger distances along the longi-tudinal direction. This results in a strong quantization along the perpendiculardirection and a weak quantization along the longitudinal direction. The electronwavenumber k‖ is effectively continuous on the scale of k⊥. The continuum ofthese k‖ states in each k⊥ mode are called 1D subbands. In Fig. 2.1c,d the 1Dsubbands are indicated by the red lines. Of course subbands only appear if thequantization energy is above kBT .

There are always two quantization lines that are the nearest to a charge neutralitypoint, one at K and one at K′. These lines correspond to k⊥ and −k⊥, indicat-ing that (in the absence of symmetry breaking) electrons can move clockwise oranticlockwise at the same energy. This is referred to as valley-degeneracy. Intu-itively one can say that electrons in the K and K′ valleys encircle the CNT withopposing 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 withspin-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 fieldOne way to break the symmetry is by applying a magnetic field. As is shown inEq. (2.2) a magnetic field parallel to the CNT axis will change the quantizationof 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 K′ and negativeat K. Since the dispersion relation is linear, we can write ΔE = �vFΔk, whichyields the shift of energy that is due to the magnetic field:

ΔE =evFD

4B‖ = μorbB‖ . (2.3)

18


Here we have defined μorb ≈ D [nm]0.2meV/T as the orbital magnetic momentthat is associated with an electron encircling a CNT of diameter D. This showsthat energy levels of states in a CNT shift when a magnetic field is applied parallelto the CNT axis. The convention is that states that move up in energy are labeledcounterclockwise (ccw) and states that move down in energy are labeled clockwise(cw). This is shown in Fig. 2.1f. Note that for typical CNTs μorb is much largerthan the Bohr magneton (μB ≈ 0.058meV/T).

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

The atomic spin-orbit coupling of a free carbon atom is Δ ∼ 12meV. This isrelatively low, because carbon has a small mass. In graphene the intrinsic spin-orbit splitting close to the charge neutrality points is even smaller (∼ 1μeV) dueto symmetry reasons. In CNTs there is an enhanced spin-orbit interaction closeto the charge neutrality points, due to their curvature. On a curved surface pzorbitals 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 toa px-up orbital on atom B (these are perpendicular to pz orbitals) and stay herefor a while before it is flipped by the (weak) atomic spin-orbit interaction, to thepz-down orbital on atom B. This process depends on the curvature of the CNTand results in a diameter depended spin-orbit interaction.

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

Δcurv ∼ 1.6meV/D [nm] . (2.4)

The spin-orbit interaction modifies the circumferential quantization conditionsand can be studied by tracking the magnetic field dependence of states in aquantum dot.

Carbon nanotube bandgapAt zero field the k⊥ spacing is determined by the diameter of the CNT. Ata particular spacing the quantization lines cut the Dirac cones exactly at thecharge neutrality points. In this case the dispersion relation is linear and theCNT has no bandgap (similar to graphene). Such CNTs are called metallic. Inpractice curvature, strain or twist can induce a small bandgap on the order of10 . . . 50meV 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 relationwith a large bandgap Eg that depends on the diameter in the following way [10]:

Eg ≈ 0.8

D [nm]eV . (2.5)

The bandgap can be changed with a magnetic field, as was described in Eq. (2.3).

19


Electron-wave quantization by longitudinal confinementThe 1D subbands are quantized by the longitudinal confinement. This leads tothe following boundary condition:

LkFE

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 semiconductingnanotubes the dispersion relation is parabolic, which results in a smaller levelspacing [11]. Due to its large Fermi velocity, a typical CNT with a length on theorder of a few 100 nm, can already have a level spacing that can be observed attemperatures of a few Kelvin. Numerous experimental studies of quantum dotsin CNTs have been reported. As we already mentioned in Sec. 1.3.3, one of themain motivations for this research is to use the spin degree of freedom of isolatedelectrons as a resource for quantum computation [12].

Shell filling in CNT quantum dotsShell filling is a characteristic feature of CNT quantum dots [13]. We will use itto experimentally identify the number of CNTs across a trench, and to determinethe 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 tothe longitudinal confinement, only discrete wavenumbers are allowed, that areindicated by the vertical dashed lines. Electrons and holes can occupy states thatreside in small bands at the points where the vertical lines cross the parabolicdispersion. These bands are indicated by the short horizontal lines. The statesin 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 offreedom. The charging energy is the electrostatic energy that is needed to adda single electron to the CNT. A single orbital level, or shell, can contain fourelectrons. 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 conductanceat 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 statebecomes available (in the energy window that is set by the applied bias), electronsor holes can tunnel on/off the quantum dot from/to the leads. In panel (c) shellfilling is shown, measured in four different CNTs, with different bandgap Eg. Thetop curve is for a small bandgap device, the bottom curve for a large bandgapdevice.

The effects of tunnel coupling and bandgap on transport are clearly visible. In thetop curve the dot is in the strong coupling regime where Ueff ≈ 0, and transportis dominated by Fabry-Perot interference (we will discuss this in Sec. 2.1.3). In

20


the lower curve the dot is in the weak coupling regime, where Ueff � 0. In thisregime the charge on the dot is quantized and transport is dominated by Coulombblockade. In this experiment the electrons in small bandgap nanotubes are weaklyconfined (small tunnel barrier) and in large bandgap nanotubes they are stronglyconfined (large tunnel barrier). In Sec. 2.1.2 we will discuss (in a simplifiedmodel) how the CNT bandgap determines tunnel coupling of a quantum dot tothe leads.

50

40

30-40 0 40

k(106 m-1)

E(m

V)

G(e

2 /h)

0

2

4

CB

FP

Ueff

Egincreasing

0 0.5 1Vg(a.u.)

Ueff+ n

Ueff

K’K’KK

(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 thelongitudinal direction (kn = nπ/L) is indicated by the evenly spaced vertical lines.Each kn gives rise to a shell consisting of four states corresponding to K, K′, spin-upand spin-down. The shells are indicated by the horizontal lines. (b) Each shell canbe 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 thedot. (c) Shell filling is apparent in conductance vs. gate voltage traces. Here are tracesshown 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 theCNT 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 bythe naturally occurring barrier at the interface of the nanotube with the contactmetal [14]. In this section we will discuss the nature of the metal-nanotubeinterface, because this will be important in choosing the contact metal for ourdevices.

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

21


In a simple approximation the Schottky barrier height ΦSB is defined for holes/electronsin the following way:

ΦSB,h = χ+ Eg − φm , (2.8)

ΦSB,e = φm − χ . (2.9)

Here is φm the metal work function and χ the electron affinity. It is common todefine the nanotube work function as φCNT = χ+ Eg/2 [16]. The expression forSchottky barriers becomes:

ΦSB,h = φCNT + Eg/2− φm , (2.10)

ΦSB,e = Eg/2 + φm − φCNT . (2.11)

Small bandgap nanotubes are favorable to achieve small Schottky barriers. Witha small bandgap CNT, three types of contact can be distinguished: Schottkybarrier, Ohmic contact or PN/NP junctions. When φCNT−Eg/2 < φm < φCNT+Eg/2 (metal work function in the gap) a barrier develops for electrons and holes.Ohmic contacts can be achieved for electrons: φm � φCNT (ΦSB < 0) and forholes: φm � φCNT. When there is an ohmic contact for electrons (holes), there isa NP (PN) junction for holes (electrons). In Fig. 2.3 we show the correspondingband alignment.

PN/NP

elec

tron

sho

les

Ohmic Schottky(a) (b) (c)

(d) (e) (f)

Figure 2.3: Bandalignment as a function of doping and metal work function.Solid lines show the CNT band bending. The Fermi energy is denoted with the dashedline. Grey color shows the band filling. (a,b,c) The metal work function and CNTdoping are decreasing from left to right. Figures show examples of Ohmic, Schottkyand PN junction. (d,e,f) The metal work function and CNT doping are increasingfrom left to right.

The width of a barrier (depletion width) can be controlled by gating the nan-otube. This pulls the bands down for electron doping (pushes up for holes),which decreases the depletion width. The barrier height can be to some extendengineered by adjusting the work function of the source drain contact. In Fig. 2.4the dependence of barrier height on work function and bandgap is illustrated.

Typical small bandgap CNTs have a bandgap of ∼ 50meV. In our experi-ment we have chosen rhenium contacts. With φRe = 4.7 eV and taking φCNT =

22


4.9± 0.1 eV, the Schottky barrier for holes/electrons is estimated to be ΦSB,h =225± 100meV and ΦSB,e = −175± 100meV. In this particular case there is anOhmic contact for electrons and a PN junction for holes. A similar situation waspreviously reported with Cr/Au contacts [17]. Large work function metals suchas Pd or Pt, usually give Ohmic contacts to holes, but bad contacts to electrons.

Eg,1>Eg,2 m,1< m,2 Eg,1>Eg,2 m,1< m,2

hole doping electron doping

Eg,1 Eg,2

SB,2

SB,1 EF

legend (a) (b) (c) (d)

Figure 2.4: Schottky barrier height of metal/CNT/metal junctions. (a,b)The Fermi level is tuned such that the CNT is hole doped. (c,d) The Fermi level istuned such that the CNT is electron doped. (a,c) Band diagrams are drawn for twonanotubes with different bandgap. (b,d) Band diagrams are drawn for two metalswith different work function. The Schottky barrier height is depicted by arrows left ofeach band diagram. For simplicity the Fermi level is taken at the same distance fromvalence/conductance band for hole and electron doping.

2.1.3 Quantum dots in the Fabry-Perot regime

In a coherent conductor the phase relaxation length is much larger than its size.In this case the conductance can be described by evaluation of the probabilitythat electrons pass through a number of channels. In the Landauer formalismthis is done by relating incoming and outgoing electron wave amplitudes throughscattering matrices. From these matrices a set of real eigenvalues can be ex-tracted, that correspond to transmission probabilities Tn(E) for each channel.Here T can have values between 0 and 1. With these transmission probabilitiesthe conductance can be expressed as:

G = e2/h∑n

Tn . (2.12)

This equation implies that even a single-channel ballistic conductor with T =1 and perfect Ohmic contacts has a finite resistance of: h/e2 = 25.8 kΩ. Ananotube has four channels due to spin and valley degeneracy and its maximumconductance is 4e2/h, corresponding to 6.45 kΩ.

In the ballistic and phase-coherent regime, the conductance through a CNT sec-tion defined by two tunnel barriers is determined by the coherent interferenceof electron waves bouncing back and forth. This type of electron transmissionis different from classical transmission, because it is phase coherent. It is calledFabry-Perot interference, in analogy to the optical Fabry-Perot etalon. The quan-tum dots that forms naturally in our devices are such cavities. The cavity mirrorsare formed by two Schottky barriers that naturally occur on the CNT/contactinterface. In Fig. 2.5 we present a schematic picture of a CNT quantum dot.

23


Figure 2.5: Schematic diagram of a carbon nanotube quantum dot. Twometal electrodes, source (S) and drain (D) are connected to a CNT. The quantum dotis formed in the segment with length L between the electrodes. The CNT is capacitivelycoupled to a gate electrode (usually the back gate plane of a silicon substrate). Theinset shows a mode of an electron wave in the CNT. Figure adapted from Ref. [18].

To find out the dependence of the transmission on phase, the nanotube quantumdot can be modeled as a double barrier system. By summation of the transmissionand reflection probabilities it can be found (see for example Ref. [19]) that thetransmission through a single channel is given by:

T =TLTR

1− 2√RLRR cosϕ+RLRR

. (2.13)

Here ϕ is the phase acquired by an electron traveling back and forth between thetwo tunnel barriers. The transmission/reflection probability of the left and righttunnel barrier are characterized by TL,R and RL,R where R = 1− T . Maximumtransmission is found when the accumulated phase during a round trip is aninteger multiple of 2π:

ϕ = 2kFL = 2nπ . (2.14)

Here kF is the Fermi wave number: kF = 2π/λF, and L is the distance betweentwo tunnel barriers. Experimentally kF can be changed with a gate voltage(eαVg), and a bias voltage (eVb). Here α is the gate coupling factor. Note thatthe level spacing ΔE corresponds to the eVb that has to be applied to accumulate2π during one round trip.

A change of gate voltage results in a change of the charge density of the nanotube.This results in a shift in Fermi energy that changes the Fermi wave numbers(momentum) kF of the electrons in the nanotube. This amounts to a phaseΔkFL. A change of source drain voltage results in a change of energy of theelectrons entering the quantum dot. This also results in a change of momentumand amounts to a phase that is due to electron motion, eV L/�vF. The result isthat the transmission and hence the conductance will oscillate as a function ofgate and bias voltage. The period of the modulation is a measure for L. Themodulation depth is determined by the transmission coefficients, as is plotted inFig. 2.6.

24


- 0 20

0.5

1

T

Figure 2.6: Transmission through a double barrier system. The transmissionof a single conductance channel between two tunnel barriers depends on the barriertransparency and the accumulated phase during a round trip of an electron travel-ing back and forth in the channel. For simplicity symmetric barriers are considered,TL = TR = 0.01, 0.2, 0.4, 0.6, 0.8, 0.99. The modulation depth of the total transmissiondepends strongly on the transmission of the tunnel barriers.

The following simple expression for the phase dependence can be found [20]:

ϕ =2π

λFL = kFL =

eVb + eαVg

�vFL+ kF0L . (2.15)

Here kF0 is the initial Fermi-wavenumber. The combination of these interfer-ence patterns show up in a checkerboard pattern that is regularly observed inCNTs [20, 21]. Regular interference patterns indicate the absence of scatteringcenters and are a signature of ballistic transport.

Tunnelrate ΓIn quantum dots with Coulomb blockade (TL,R � 1), the tunnelrate Γ can befound by fitting a Lorentzian to a Coulomb peak. This is not possible in theFabry-Perot regime where TL,R � 1. In this case Γ can be found by consideringthe probability of electrons passing through the barrier.

Once on the CNT, it takes an electron a time 2τ = 2L/vF to travel back and forthbetween the two barriers. During that time it has a probability of TR(1 − TL)to leave the dot. The probability for electrons to leave the dot is then ΓR =TR(1−TL)/2τ . In a similar way we find that the probability for electrons to enterthe dot is ΓL = TL(1−TR)/2τ . When ΓL �= ΓR we find Γ = (ΓL ·ΓR)/(ΓL+ΓR)).This analytical expression can be fitted to a conductance peak, and yields ΓL,R

and TL,R.

2.1.4 Quantum dot- superconductor coupling regimes

Transport through carbon nanotube-superconductor interfaces is determined bythe strength of the coupling between the CNT and the superconductor [22]. Threedifferent regimes can be considered, depending on the relative values of tunnel-coupling: hΓ, the (proximity induced) superconducting gap: Δ, and the chargingenergy: U . Next we list the different regimes and point out the dominant trans-port physics.

25


� Strong coupling, hΓ � Δ, U .Quantum interference of non-interacting electrons, Fabry-Perot interfer-ence.

� Weak coupling, hΓ � Δ, U .Charging effects dominate transport, Coulomb blockade.

� Intermediate coupling, hΓ ∼ Δ ∼ U .Charging effects important, but higher-order tunneling process significanttoo, cotunneling and Kondo effect.

Strong coupling ( U

Weak coupling ( U

Intermediate coupling ( U

(b)

U E+D

0

VG

S

S

QD S

(a)

(c) (d)

(e)

Figure 2.7: Coupling regimes and characteristic energy scales. (a) Schematicdevice geometry. The quantum dot (QD) is formed on an isolated island connected totwo superconducting electrodes (S). The QD is capacitively coupled to a gate electrode.(c) Energy diagram of the device shown in (a). The superconductor quasi-particle DOSis depicted in light/darkgrey. The tunnel barriers are depicted by the vertical bars.The superconducting gap is indicated by Δ, the chemical potential in the source/draincontacts by μS,D. Γ indicates the level broadening of states on the QD that are spacedby the charging energy U and ΔE. The detuning of the lower state with μD is indicatedby ε0. (b,d,e) Different coupling regimes are shown. In the strong coupling regime,Cooper pair transport is possible. In the weak coupling regime transport is dominatedby single-electron tunneling. In the intermediate coupling regime, Cooper pair transportcan take place via an indirect process. Figure adapted from Ref. [22].

In the weak coupling regime, the charge on the QD is quantized and electronshop on and of the CNT one by one. When a bias voltage larger than 2Δ/e isapplied, single electrons can tunnel from filled quasiparticle states in the sourceto empty quasiparticle states in the drain. In Fig. 2.7b the energy diagram andquasiparticle DOS is shown. When |V | < 2Δ/e there is no “direct” transportpossible. In the weak coupling regime, transport of Cooper pairs through the

26

2.2. Andreev reflection and supercurrent

quantum dot is possible via co-tunneling processes. In the strong coupling regimeCooper pair transport is possible through Andreev bound states (ABS). In thefollowing section the origin of ABS, and their consequence (supercurrent), willbe discussed.

2.2 Andreev reflection and supercurrent

2.2.1 Proximity induced superconductivity

When a normal metal is connected to a superconductor only, the electrons in themetal pick up correlations from the superconductor. This results in a mini-gapin the normal metal DOS. The size of the mini-gap depends on the timescale τ ittakes for an electron to reach the superconductor, which is characterized by theThouless energy: ETh � �/τ . For ballistic systems it is:

ETh,b = �vF/ξ = 2Δ , (2.16)

where Δ is the proximity induced gap and ξ is the lengthscale on which cor-relations are picked up, which can be larger than the size of the N part of thejunction. For diffusive systems the Thouless energy is:

ETh,d = �D/L2 , (2.17)

where D is the diffusion constant.

The mini-gap decays on a length scale ξ, the superconducting coherence length.In the case of ballistic transport, ξ is related to the Fermi velocity and the su-perconductor gap in the following way:

ξb =�vF2Δ

. (2.18)

And in a diffusive system:

ξd =

√�D

2Δ. (2.19)

The CNTs discussed in this thesis are in the ballistic regime and can be stronglycoupled to the superconductors. In such junctions proximity induced supercon-ductivity appears. This originates from transport of Cooper pairs by Andreevreflection, which is the topic of the remaining part of this section. In Fig. 2.8awe give a schematic picture of the proximity effect.

2.2.2 Andreev reflection

Transport of charge through a Josephson junction is described by similar scat-tering matrices that were discussed in the first part of Sec. 2.1.3. However, in thepresence of superconductivity an extra contribution to the transport that origi-nates from Andreev reflection has to be considered. We will first discuss Andreev

27


Figure 2.8: Proximity induced superconductivity. (a) The order parameter Ψdecays on a lengthscale ξ at the interface of a superconductor to a normal metal. A finitedensity of Cooper pairs is found in the normal metal. (b) When two superconductorsare coupled by a CNT weak link, Cooper pairs can leak in the CNT. When the contactspacing L � ξ, where ξ is the coherence length of Cooper pairs in the CNT, transportof Cooper pairs between source and drain is possible. Figure adapted from Ref. [23].

reflection (AR) at a single N/S interface, and secondly the formation of ABS inS/N/S junctions. These are the states that carry supercurrent.

An electron at an energy E < Δ that is traveling from the normal metal andimpinging on the superconductor cannot propagate in the superconductor, be-cause there is a gap in the single particle density of states. As a consequencethe electron wave is exponentially damped in the S region, implying that it hasto be reflected. It turns out that the only possible way that the electron, with

momentum−→k , can enter the superconductor is by pairing with a second electron

with momentum −−→k , inside the normal region, and form a Cooper pair that joins

the superconducting condensate. What is reflected then is the missing −−→k elec-

tron. This process can be alternatively described as the reflection of a hole-like

quasiparticle (having momentum ≈ −→k ) moving away from the superconductor.

In Fig. 2.9 this process is schematically shown.

e

h

Cooper pair

i(ϕe+kex)ψe=e

i(ϕh-khx)ψh=e

E

N S

Ef

Δeiϕs

Figure 2.9: Andreev reflection on a S/N interface. An electron momentum keand energy E < Δ is impinging on a superconductor, described by Δeiϕs . The electronis reflected as a hole with momentum kh. A Cooper pair is added to the superconductingcondensate. Figure adapted from Ref. [24].

This process is the only process compatible with the conservation of energy andmomentum (which is almost conserved) in the normal metal. The momentumof the reflected hole is almost equal to that of the impinging electron: �kh =�ke − 2E/vF, the difference 2E/vF is due to the force that acts on the electron

28


in the superconductor, just before it is reflected. This force is too small to

reflect the electron as an electron, because for that a momentum change of 2−→k

is needed. Since the kinetic energy EF is much larger than the energy providedby the superconductor Δ, (Δ/EF is typically 5 eV/1meV) the superconductorcannot provide the energy to reverse the direction of the electron. Reflection as aparticle with velocity in the direction opposite to its momentum is the only wayout, electrons are reflected as holes. From the viewpoint of the normal metal,AR results in a charge deficit of 2e [25, 26].

The scattering theory for AR is described by the BTK theory [27]. This theorypredicts two important consequences. In the case of pure ballistic transport,and a perfectly transparent interface (no tunnel barrier, strong coupling), theconductance measured at low bias (V < Δ/e) is twice as high as that measuredat high bias V > Δ/e. This is the direct consequence of 2e charge transport dueto AR. In the case when the S/N junction is formed by a tunnel barrier (weakcoupling), AR is suppressed and in this case the low bias conductance is muchsmaller than the high bias conductance.

2.2.3 Andreev bound states

In an S/N/S junction (in the strong coupling regime) AR happens continuouslyon both interfaces. An electron impinging on the surface of the superconductorwill be reflected as a hole. As the reverse process of e − h reflection can alsooccur, the hole will impinge at the surface of the other superconductor, and isreflected as an electron. This leads to a coherent circulation of electrons in onedirection and holes in the other. This is schematically represented in Fig. 2.10.

E

h

e

N

Cooper pair Cooper pair

S2

Ef

S1Δe

iϕs1Δe

iϕs2

L

Figure 2.10: Andreev bound states in an S/N/S junction. Coherent circulationof electrons and holes confined in a normal region between two superconductors resultsin the formation of ABS. In every cycle, one Cooper pair with charge of 2e is transportedfrom S1 to S2. The opposite process is also possible, in that case electrons move to theleft and holes move to the right, resulting in a current from S2 to S1. Figure adaptedfrom Ref. [24].

Just as was the case in the discussion of the Fabry-Perot regime in Sec. 2.1.3,electron-hole waves bouncing back and forth between the superconductors caninterfere constructively when the total acquired phase during one cycle is a mul-tiple of 2π. This results in a discrete number of bound states that are calledAndreev bound states.

29


The total acquired phase is the sum of two terms. The first term is the phasepicked up during AR, and the second term is the phase that is picked up due toelectron motion. In the discussion of the Fabry-Perot regime we already encoun-tered the dynamical phase that is due to electron motion LE/�vF = kFLE/EF.The phase that is picked up during AR can be derived from the Bogoliubov deGennes equation (see for example Ref. [28]) and is ϕ = −2 arccos (E/Δ)± (ϕ1 −ϕ2). Here ϕ1,2 are the phases of the superconducting condensates in the twosuperconductors, and ± corresponds to the two possible directions in which theelectron-hole pair can circulate. The resonance condition is now:

−2 arccos

(E

Δ

)± (ϕ1 − ϕ2) + kFL

E

EF= 2πm . (2.20)

Here m is equal to the number of ABS that can exist in a junction of size L. Tworegimes can be considered depending on the relative contributions of the firstand last term in Eq. (2.20). In a “long” junction the phase accumulation due tothe dynamical phase is dominant (in this regime, L � ξ). In a “short” junctionthe phase accumulation is essentially only due to phase picked up during the ARprocess. It can be derived that in a short one-dimensional ballistic junction twodoubly degenerate ABS appear in each transport channel [28], with its energygiven by:

En = ±Δ

√1− Tn sin

2(ϕ/2) . (2.21)

Here Tn is the transmission coefficient for the channel n, and ϕ = ϕ1 − ϕ2 isthe phase difference across the junction. The energy spectrum of the ABS isplotted in Fig. 2.11. For junctions with small transmission (a tunnel junctionwith T � 1), the energy phase relation is a cosinusoidal just below the gap edge.For junctions with large transmission (a quantum point contact with T � 1) it hasa cusp at ϕ = π. It should be noted that each ABS come in pairs, correspondingto left and right circulating e− h trajectories with opposite energies. In the caseϕ = 0, each ABS is doubly degenerate. Normally a Josephson junction is in itsground state, and its energy dispersion is the negative counterpart of the curvesplotted in Fig. 2.11.

2.2.4 Supercurrent

Andreev bound states carry supercurrent. This can be shown by considering thephase dependence of all the ABS in a junction. The total phase dependent energyis the sum of all ABS:

E(ϕ) =∑n

En(ϕ) = Δ∑n

√1− Tn sin

2(ϕ/2) . (2.22)

It can be shown that the supercurrent in the junction is proportional to ∂En/∂ϕ [28],which we can use to calculate the supercurrent:

I(ϕ) = −2e

�

∑n

∂En

∂ϕ=

eΔ

2�

∑n

Tn sin(ϕ)√1− Tn sin

2(ϕ/2). (2.23)

30


0 2

0.1

1

TransmissionI(e/2

)E/

0

1

1

1

0

Figure 2.11: Energy dispersion of ABS. (top) The energy of an ABS dependson the phase difference ϕ between the order parameters of the two superconductorsbetween which the ABS is confined. It also depends on the transmission coefficientsat the boundaries. For junctions with small transmission (T � 1), E has a cosinedependence on phase (upper curves), for junctions with large transmission T � 1, thedispersion has sharp cusps at −π and π (lower curves). (bottom) Correspondingsupercurrent. The current phase relation is only sinusoidal for junctions with T � 1.For such junctions, the maximum supercurrent is found at ϕ = π/2. For junctions withT � 1 the current phase relation is highly non-sinusoidal and the maximum current isfound at ϕ = π. Figure adapted from Ref. [29].

In Fig. 2.11 we plotted I(ϕ) for several different transmission coefficients.

Historically the first Josephson junctions were tunnel junctions in the regime of

T � 1. In this regime, I(ϕ) = Ic sin(ϕ), with Ic = (eΔ/2�)∑n

Tn = (πΔ/2e)Gn,

where Gn is the normal state conductance of the junction. Maximum super-current is achieved at ϕ = π/2. The Josephson energy is defined by: EJ =−EJ0 cos(ϕ), with EJ0 = �Ic/2e.

The Josephson effect and RCSJ model (Sec. 2.3) are usually described in termsof these energy/current phase relations. We have seen that these depend heavilyon the transmission coefficients of the junction. As will be discussed in Sec. 5.3,CNT Josephson junctions can have transmission coefficients T � 1, and hencewe expect that the energy/current phase relations can deviate strongly fromsinusoidal relations. For example in the case of perfect transmission T = 1,I(ϕ) = Ic sin(ϕ/2), with Ic = (πΔ/e)Gn. The current-phase relation approachesa linear phase dependence (saw-tooth like) where maximum supercurrent is foundat ϕ = π. Non-sinusoidal current phase relations in CNTs have not been directlymeasured [29] and are an ongoing topic of study. Indirect evidence for non-sinusoidal behavior has been found in CNT Josephson junctions [30, 31].

31


2.3 Josephson effect

In this section we will review the properties of Josephson junctions. We willfirst discuss the DC Josephson effect and the RCSJ model, and then discuss theAC Josephson effect. Since many good descriptions of this topic can be foundin textbooks we have chosen to closely follow them here. Our main sources areRefs. [32–35]. Throughout we neglect effects of finite temperature.

2.3.1 General description

The Josephson effect occurs in weak links coupling two superconductors. Twosuperconductors are considered to be weakly linked if the wavefunctions of thesuperconductors can overlap. The current in the weak link between two bulkysuperconducting electrodes may contain a supercurrent component Is, which isnot depended on the voltage V across the electrodes but on the phase difference:

ϕ = ϕ2 − ϕ1 . (2.24)

Here ϕ1 and ϕ2 are the phases of the macroscopic Ginzburg-Landau order pa-rameter Δ in the electrodes,

Δ1,2 = |Δ1,2| exp(iϕ1,2) . (2.25)

Josephson demonstrated that the current phase relation Is(ϕ) can have a simplesinusoidal form [36]:

Is = Ic sin(ϕ) ,where Ic = (2e/�)EJ0 . (2.26)

This is a specific case of a current phase relation. In principle each Joseph-son junction has a unique current phase relation, which can deviate from thissimple case. For example, theory on 1D junctions predicts forms of Is(ϕ) thatstrongly deviate from simple sinusoids [32]. The sinusoidal current phase relationEq. (2.26) is referred to as the first Josephson relation. The second Josephsonrelation relates the phase ϕ to a voltage V across the leads:

dϕ

dt=

2e

�V . (2.27)

A constant voltage results in a phase that is linearly increasing in time. In turn,an increasing phase results in an oscillating supercurrent. This relation holds forall Josephson junctions (also those with non-sinusoidal I(ϕ) relations) and yieldsa precise correspondence between voltage and frequency:

VJ =h

2ef , 1μV ↔ 483.6MHz . (2.28)

Josephson junctions in CNTs usually have small switching currents on the orderof 1 nA [22]. For this reason filtering is necessary to observe the Josephson effect,and this limits the bandwidth of the experimental setup. This makes it techno-logically challenging to directly measure oscillating supercurrent. For voltages

32

2.3. Josephson effect

above kBT in a dilution fridge (∼ 2μeV) at 20mK, frequencies are in the rangeof 1 . . . 10GHz.

As we will show in Sec. 2.3.3, there is another method to probe the AC Josephsondynamics. This involves the study of Shapiro steps [37]. These are equidistantplateaus that appear on IV curves, and are measured in a current bias configu-ration.

Josephson junction IV curves are not only determined by the properties of theJosephson junction, but by the full measurement circuit. A convenient model forunderstanding of IV characteristics of a current biased junction and its dynamics,is the resistively and capacitively shunted junction (RCSJ) model [38].

2.3.2 The RCSJ model

In the RCSJ model the Josephson junction is modeled by a Josephson elementwith I = Ic sin(ϕ) and uses Eq. (2.27) and standard network theory to find thetime averaged voltage and current behavior. In this section we will discuss theRCSJ model and the intrinsic properties of Josephson junctions. The modelcircuit is shown in Fig. 2.12.

IC sin( )CR IV (t)

Figure 2.12: Circuit used in the RCSJ model. The cross depicts the Josephsonelement.

By application of Kirchhoff’s rule and substitution of Eq. (2.27), the equation ofmotion for the phase of the circuit is derived:

C�

2eϕ+

1

R

�

2eϕ+ Ic sin(ϕ) = I(t) . (2.29)

This equation is similar to the equation of motion of a particle moving alongthe ϕ axis in a tilted washboard potential. In the RCSJ model we refer to thedynamics of the phase particle in the tilted washboard potential (Fig. 2.13). Theeffective potential is given by:

U(ϕ) = −EJ cos(ϕ)− (�I/2e)ϕ . (2.30)

The particle experiences a viscous drag force Fdrag = (�/2e)2(1/R)dϕ/dt, and hasa mass m = (�/2e)2C. The resistor damps the dynamics of the phase particle.A junction with large C (mass) is harder to damp then a junction with small C.The dynamics of the phase-particle ϕ are characterized by two frequency scales:The plasma frequency ωp =

√(2e/�)Ic/C, and the characteristic frequency ωc =

(2e/�)IcRn.

33


0 2 3 4 5

0

U (a

.u.) I=0

I=0.5IC

I=IC

I=1.5IC

I=2IC

/4

Figure 2.13: The tilted washboard potential representation of the RCSJmodel. Black dots indicate the location of the phase particle. The potential can betilted by applying a bias current I. By tilting the potential, the phase particle moves tothe right. When the phase particle has moved by π/4, the particle can hop through thewashboard potential (this corresponds to I > Ic). The particle moves down, generatingan (almost) DC voltage proportional to its phase velocity.

Equation (2.30) can be rewritten in terms of ωc and ωp:

1

ω2p

ϕ+1

ωcϕ+ sin(ϕ) =

I(t)

Ic. (2.31)

Next we will discuss the intrinsic parameters and properties of a Josephson junc-tion in the RCSJ model.

Josephson plasma frequencyThe plasma frequency:

ωp =√

(2e/�)Ic/C , (2.32)

is the resonance frequency at which the equivalent of potential and kinetic energyoscillate with π phase difference.

Potential energyIn a Josephson junction potential energy can be stored by a phase winding ofΔϕ. By increasing the bias current I, the phase winding and hence the potentialenergy is increased.

Kinetic energyThe charging energy of the RCSJ circuit is the equivalent of kinetic energy. Acharge imbalance Δq = 2ne is induced by the transfer of n Cooper pairs. Thisincreases the electrostatic energy by the charging energy: Ec = Δq2/2C.

The potential energy oscillates as EJ(1−cos(ϕ)) and the charging energy oscillatesout of phase with (2ne)2/(2C). The Hamiltonian of this system is similar to thatof a pendulum. The system has a resonance frequency (plasma frequency) that is

proportional to√EJ/C, which can be expressed in terms of Ic as is done above.

34


Characteristic frequencyThe characteristic frequency:

ωc = (2e/�)IcRn , (2.33)

is the rate of the phase winding when I = Ic. In this case there is a characteristicvoltage Vc = IcRn across the junction. At this point on the IV curve it isinteresting to compare the charging time of the RCSJ circuit τ = RC to the rateof phase winding ωc = 2e/�Vc. These dynamics are characterized by the qualityfactor of the circuit.

Quality factorThe number of oscillations at ωc within τ = RC are given by the McCumberparameter β = ωcRC. It is common to express this parameter in terms of theplasma frequency ωp, which yields the quality factor Q of the circuit:

Q =√

β = ωpRC . (2.34)

Both Q and β are a measure for the damping of the phase particle. Two casesare considered: 1. Q < 1, the junction is overdamped, 2. Q > 1, the junctionis underdamped. In the first case, heavy damping implies that the viscous dragforce Fdrag is much larger than the inertia of the phase particle. The velocity ofthe particle is proportional to the local slope of the washboard. In this case theIV curve is single valued and is a smooth interpolation from V = 0 to V = IR.In case two the inertia is dominant, and the running phase particle has enoughkinetic energy to overcome a number (β) of washboard oscillations before it isretrapped. This effect results in a double valued IV curve and leads to hysteresisin the observed switching current. The retrapping current in an ideal tunneljunction is inversely proportional to Q:

Ir/Ic ≈ 4/(πQ) . (2.35)

2.3.3 Shapiro steps on IV curves

The dynamics of a Josephson junction can be studied in a DC measurementconfiguration by the study of Shapiro steps [37]. They manifest themselves ashorizontal plateaus that appear on an IV curve of a current biased junction thatis irradiated by microwaves. The Shapiro steps are a result of the time-averagedphase dynamics. The plateaus are equally spaced by a voltage which correspondsto the applied microwave frequency via Eq. (2.27). The width of the Shapiroplateaus (region of supercurrent in between two steps) has a very characteristicdependence on microwave power. In this section we will give a simple picture onthe origin of Shapiro steps. First we will show how the phase locking between thetime evolution of the phase particle and the microwave signal leads to Shapirosteps and secondly we will show how the Shapiro plateau width depends on power.

Phase lockingPhase locking between the time evolution of the supercurrent and the RF mi-crowave signal results in a time averaged supercurrent that can be larger thanzero, while a DC voltage is maintained across the junction. Because phase locking

35


is only possible at integer multiples of the frequency of the microwave signal ω1,the voltage across a current biased junction can only exist at integer multiples ofthe Josephson voltage at this frequency:

Vn = n�ω1

2e. (2.36)

A graphical representation of phase locking is given in Fig. 2.14.

0 5IC 0 -IC IC 0 -IC

V5=5V1 V4=4V1

V3=3V1

V2=2V1

V1

0

8

16

24

t (2 J)-1

/4

IC 0 -IC

(a) (b) (c) (d)

Figure 2.14: Time-averaged phase dynamics lead to Shapiro steps in IVcurves. In (a),(b) and (c) the current phase relation is plotted in three differentscenarios. In panels (a) and (b) the period of the phase dynamics is an integer multipleof the period of the current phase relation. In this case the time evolution of thesupercurrent is phase locked to the RF signal. In panel (c) the time evolution of thesupercurrent is not phase locked to the RF signal. In panel (d) the phase dynamicsof a voltage biased junction irradiated by an RF signal is shown. The phase evolutionis not constant in time. The black dots denote the regions where the phase particlespends the most time and where the contribution to the time-averaged current is thelargest. Only the phase locked time evolution leads to a time-averaged supercurrentthat can be nonzero. This is denoted by the grey bands in panels (a) and (b). Thisshows that the junction can be in a supercurrent carrying state, while maintaining aDC voltage across it. The voltage can only take discrete values on integer multiples ofthe Josephson voltage.

36


Power dependence of Shapiro stepsShapiro steps have a very characteristic power dependence which can be derivedfrom the time-average of the oscillating supercurrents. On a Shapiro plateau thetotal voltage is: V = V0 + V1 cos(ω1t). This results in a phase-evolution:

ϕ(t) = ϕ0 + ω0t+ (2eV1/�ω1) . (2.37)

Here ϕ0 is a constant phase offset which can be for instance applied by a currentbias, ω0 = 2eV0/� is the linear phase increase due to the time average of thevoltage on the plateau, and the last term is an oscillating phase term. Theexpression for Is(t) is found by inserting Eq. (2.37) in Eq. (2.26) and expandingthe sin(sin(ϕ)) term in Bessel functions [39]:

Is(t) = Ic

∞∑−∞

(−1)nJn(2eV1/�ω1) sin(γ0 + ω0t− nω1t) . (2.38)

A plot of Is(t, 2eV1/�ω1) at ϕ0 = π/2, on the zero-voltage Shapiro plateau isgiven in Fig. 2.15a,b.

0 5 10 15 20-0.5

0

1

I(t)

(I C)

0

10

202IC

0 2

(a)20

10

0

-IC0IC

eV1/

1

eV1/ 1

0 2

eV1/

1

1t

1t

(b)

(c)

Figure 2.15: Supercurrent oscillations on the edge of the V = 0 Shapiroplateau (ϕ0 = π/2). (a) One period of Is(t, 2eV1/�ω1) in colorscale. (b) Linetracestaken from (a) as a function of power. The faint lines indicate zero current. Linetracesare offset by 2Ic. (c) The time averaged supercurrent 〈Is〉 has a 0th order Bessel function(J0) power dependence. Black arrows indicate the first two nodes of J0 in (a,b,c).

The width of a Shapiro plateau is found by calculating the DC current biaswhich has to be applied to reach the time-averaged supercurrent where the volt-age jumps to the next plateau. At this point the tilt of the washboard po-tential is such that ϕ0 = π/2. The width has a characteristic power depen-dence that follows Bessel functions, i.e. the nth Shapiro plateau has a width ofIn = 2Ic|Jn(2eV1/�ω1)| [34].

37


Spectral density of Josephson currentThe non-linear character of the Josephson relations results in currents that os-cillate at integer and fractional multiples of the driving frequency. For a fixeddriving frequency ω1, the power spectral density (PSD) of the Josephson currentdepends on the microwave power and on the voltage across the junction. We havenumerically calculated Is(t) = Ic sin(sin(ϕ(t))) as a function of power for the firstfive Shapiro plateaus. In Fig. 2.16 we have plotted the PSD of Is(t). With in-creasing power more harmonics of the driving frequency appear. At Vn > 0 thelargest contribution comes from currents with a frequency of ωn = 2eVn/�. AtV = 0 only double integer multiples of the driving frequency appear. Recurringnodes appear in the PSD as a function of power.

20

10

0

20

10

0

20

10

00 4 288 12 16 20 24 0 4 28 328 12 16 20 24

V5=5V1V4=4V1

V3=3V1V2=2V1

V0=0 V1=( /2e) 1

PS

D(a

.u.)

1

2

3

4

0

eV1/

1eV

1/1

eV1/

1

/ 1 / 1

(a) (b)

(c) (d)

(e) (f)

Figure 2.16: Power spectral density of Josephson current as a function ofvoltage and power. All spectra are obtained by a DFT from 104 datapoints sampledat 100ω0. The time series are multiplied by a Hamming window to minimize the effectof the dataset boundaries. For better visibility the linewidth is increased by a movingaverage filter with a span of ω0/8. Black arrows indicate the first two nodes of J0 inFig. 2.15c.

38


Linewidth of Josephson currentThe second Josephson relation Eq. (2.27) implies that the frequency of the Joseph-son current changes with the voltage bias. When the applied voltage would beperfectly constant, the power spectral density of the Josephson current would beinfinitely sharp at the corresponding Josephson frequency. In practice, there isalways noise present that effectively broadens the spectrum of the Josephson cur-rent. By analysis of the RCSJ model with thermal noise, the following expressionsfor the linewidth can be found [33,40],

Γ = 4πR2

d

Rn

(2e

h

)2

kBT . (2.39)

This gives at Rd = Rn a linewidth of ∼ 40MHz per each Ohm of the junctionnormal resistance, per each Kelvin of the absolute temperature. Here Rd mea-sures the local resistance on a curve with Shapiro steps (it is low at a plateauand high on a step). This equation has been derived with Al/Al2O3/Al tunneljunctions in mind. There the normal state resistance Rn is typically defined bydefects in the aluminum-oxide tunnel barrier.

In CNT junctions, such defects do not exist and the phase dynamics are deter-mined by dissipation and temperature in the impedance of the circuit that is usedto bias the junction. Therefore it is not possible to apply this expression directlyto our junctions.

In a different approach with a shunted voltage biased junction, the thermal noiseacross the shunting resistor is used to approximate the linewidth. In that case,the linewidth is found to be [41]:

Γ = 8Rn

(2e

h

)2

kBT . (2.40)

This result yields a linewidth of ∼ 26MHz per Ohm, per Kelvin. For our CNTdevices we do not know how to predict the linewidth of supercurrent. Equa-tion (2.39) is not applicable because the nature of Rn is different in our devices,and Eq. (2.40) is not so useful because the impedance of our device is not wellknown.

The effect of a shunting resistor in a current biased junction is to smooth outIV curves. At high power, non-linearity of the circuit impedance can result inrectification of the applied RF voltage, that results in an effective DC voltageacross the junction. When the junction is current biased, this effect appears asan effective shunting. This is an explanation for the disappearance of Shapirosteps at high power. At low power it is not clear how exactly Shapiro steps inCNTs are smoothed. Temperature and circuit impedance are likely candidatesto play a role in smoothing of Shapiro steps. Following Ref. [40], the linewidthdepends on the local slope Rd. If this also holds in CNT junctions we expect anarrow linewidth at a plateau, and a broad linewidth on the step between twoplateaus.

39


2.4 Carbon nanotube mechanical resonator

In the simple model for the suspended CNT resonator the equation of motion isderived by modeling the CNT as a homogeneous cylindrical rod [42]. Althoughsingle-wall CNTs are hollow, rather than homogeneous, this model gives an es-timation of the resonance frequency that is in agreement with experimentallyfound resonance frequencies [43]. The equation of motion for the driven, non-linear resonator (Duffing resonator) is:

y + Γy + ω20y − βy3 =

F (t)

m. (2.41)

Here fr = ω0/2π is the resonance frequency, m is the mass of the resonator andΓ = ω0/Q is the damping rate. Q is the quality factor and F (t) is the drivingforce.

The resonance frequency fr can be estimated in terms of the bending springconstant EYI (EY is Young’s modulus and I is the moment of inertia) and themass [42]:

fr =22.4

2πL2

√EYI

ρS. (2.42)

Here L is the length of the suspended CNT, ρ is its density and S is the cross-sectional surface. The moment of inertia is I = πr4/4 [44]. This is the moment ofinertia of the cross-section (moment of area) of a beam. It describes its resistanceto bending and deflection, which depends on the geometry of the beam cross-section (think about a H-beam used in building construction). It is not themoment of inertia that describes the kinetics of a rotating object. We use this torewrite Eq. (2.43):

fr =22.4r

4πL2

√EY

ρ. (2.43)

Typical parameters for (single wall) CNTs are: EY = 1.25TPa, r = 1.4 nm andρ = 1350 kg/m3. In Fig. 2.17 we have plotted fr for a typical range of CNTlength. It has to be noted that Eq. (2.43) gives only an estimate of the resonancefrequency. Typical devices can deviate from this simple model and have a differentresonance frequency, due to tension for example.

The cubic nonlinearity of the resonator is described by the Duffing parameterβ. Its origin stems from the forces of elastic tension. It can be estimated tobe β ≈ ω2

0/r2 (see App. C). Within the resonant approximation |ν| ≤ Γ � ω0,

where detuning is defined as ν = ω − ω0, Eq. (2.41) can be solved in frequencyspace for the complex amplitude of motion yω:

yω =Fω

m

1

2ω0ν − β|yω|2 − iΓω0, (2.44)

where Fω denotes the Fourier component of the driving force at frequency ω.

40

2.4. Carbon nanotube mechanical resonator

200 300 4000

1

2

3

L(nm)

f r(G

Hz)

0 200 4000

10

20

30

L(nm)

f r(G

Hz)

(a) (b)

Figure 2.17: Resonance frequency and resonator amplitude. (a) Length de-pendence of resonance frequency of the fundamental bending mode, after Eq. (2.43).(b) Zoom in on the 0 . . . 3GHz region. Solid lines correspond to a CNT with a radiusof r = 1.4 nm. Dashed lines are for CNTs with a radius of r ± 0.5 nm.

The transition between the linear and nonlinear regimes occurs at a critical am-plitude, yc, which is defined by βy2c = 2ω0Γ. Substituting the estimated scale forβ we find the estimation for the critical amplitude:

y2c ≈ r2/Q . (2.45)

The driving force corresponding to yc can be estimated as Fc ≈ Mβy3c ≈ Mω20yc/Q

or, in terms of the CNT Young’s modulus:

Fc ≈ 102 EYS(r/L)3Q−3/2 . (2.46)

In terms of the defined critical parameters, Eq. (2.44) takes the form:

yωyc

=Fω

Fc

1

[2ν/Γ− 2(|yω|/yc)2]− i. (2.47)

In Fig. 2.18a we show the amplitude of motion |yω|/|yc|, as a function of drivingforce amplitude in three regimes: The linear, intermediary and nonlinear regime.

The solution of the Duffing equation (2.41) is multivalued when the resonator isdriven beyond Fc. This results in bistability of the resonator amplitude, as is clearfrom the tilting resonance peak in Fig. 2.18. Bistability implies hysteresis, andsharp jumps between stable states. This has been observed in CNT resonatorsbefore [43]. The non-linear response also results in a saturation of the resonatoramplitude at F > Fc. This is shown in Fig. 2.19a.

In the next paragraphs we will discuss the effect of saturation on the power de-pendence of transduced signals due to mixing or rectification. These transductionmechanisms will be discussed in detail in Sec. 2.5. Here we will show how thepower dependence of the measured signal allows for a distinction between mixingand rectification. For this discussion it is necessary to know that our CNTs aredriven by a force F that is proportional to an applied voltage Vrf . This forcehas a magnitude F , and the applied voltage also induces an oscillating voltagewith amplitude Vmax across the nanotube. In the linear regime |ymax| ∝ F ∝ V ,where |ymax| is the maximum amplitude of the CNT.

41


-10 0 10 20-20( /2)

0

0.5

1

1.5

|y|(

|yc|

)

2

F=Fc/2F=Fc

F=2Fc

Figure 2.18: Resonance lineshapes as a function of driving force. The contin-uous, dashed and dashed-dotted lines correspond to the linear regime, an intermediaryregime and the nonlinear regime.

0 1 2 30

0.5

1

1.5

F(Fc)

|ym

ax|(

|yc|

)

0 1 2 30

2

4

F(Fc)

Vm

ax|y

max

|

0 1 2 30

2

4

F(Fc)

|ym

ax|2

(b) (c)(a)

F F2 F2

Figure 2.19: Amplitude of oscillations as a function of driving force. Blackcurves: numerically calculated amplitude. Grey curves: linear extrapolation of ampli-tude in F < Fc regime. (a) Amplitude of non-linear resonator as a function of drivingforce. (b) Amplitude of transduced signal V |ymax| as a function of driving force, whereVmax ∝ F . (c) Amplitude of transduced signal |ymax|2 as a function of driving force.Grey curves are plotted to show the difference between linear and non-linear regime. Inthe linear regime the mixing/rectification signal has a quadratic dependence on F .

When driven with F < Fc, a non-linear resonator behaves linearly, its amplitudebeing proportional to the driving force. As a consequence the response of mixingor rectification signals is quadratic as a function of driving force (driving voltage),since mixing relies on a term with yVrf and rectification on a term with y2 (thiswill be discussed in Sec. 2.5). The situation is different when F > Fc, theresonator amplitude will stop increasing and saturate. In Fig. 2.19 we shownumerical calculation of the mixing and rectification signals as a function ofdriving force.

42

2.4. Carbon nanotube mechanical resonator

Table 2.1: Resonance frequency and amplitude as a function of length.

L(nm) 60 100 140 180 210 240 260 280 300 375f(GHz) 21 7.6 3.9 2.3 1.7 1.3 1.1 0.97 0.84 0.54|ymax|/(QF0) 4.5 20.9 57 122 192 288 369 460 567 1100

The maximum CNT amplitude on resonance can be found by consideration of thedisplacement in time domain. This can be recovered from the complex amplitudeyω by inverse Fourier transform:

y(t) = Re[yωe

iωt]

. (2.48)

From Eq. (2.48) the maximum displacement on resonance is now found:

|ymax| = QF0

mω20

. (2.49)

Where the driving force has an amplitude F0 < Fc. In Tab. 2.1 we presenta list of values for frequency and normalized amplitude. Short CNTs havehigh resonance frequencies, but smaller amplitudes for a given force and qualityfactor. In this case the displacement detector needs to have a larger sensitivityto measure high frequency vibrations for a given force compared to long CNTresonators. High Q devices have a larger displacement for a given driving force.

In a typical suspended CNT the oscillating gate force is the dominating drivingforce. The gate force is given by:

F0 =∂Cg

∂yVgVg . (2.50)

The oscillating gate voltage Vg is typically capacitively induced by a nearby RFantenna [43].

In the suspended beam approximation the free energy of the resonator is givenby [44]:

E =EYI

2L. (2.51)

Because the static gate force is the dominating force, the free energy of theresonator is given by the electrostatic energy of the CNT:

1

2CgV

2g =

EYI

2L. (2.52)

In the following section we will see how the Josephson energy adds to the totalenergy of the CNT, and discuss its effect on mechanical vibrations.

43


2.5 Vibrating suspended carbon nanotube Joseph-son junctions

Coupling of mechanical displacement to the Josephson effect in suspended nan-otubes can be either capacitive (mediated by electric fields) or inductive (me-diated by magnetic fields). With capacitive coupling nanotube vibrations aredriven by a Coulomb force, and with inductive coupling by a Lorentz force. Inour devices the Lorentz force is expected to be much weaker then the Coulombforce (see App. C), and for this reason we only discuss capacitive coupling. Firstwe will describe drive and detection methods on suspended CNTs with normalcontacts, and secondly we will discuss drive and detection in suspended CNTswith superconducting contacts. Here we foresee the possibility of an extra (ca-pacitive) driving force that is due to the Josephson energy.

2.5.1 Capacitive coupling in a suspended CNT with normalcontacts

Driving: Coulomb forceA suspended CNT is capacitively coupled to all conductors in its vicinity. Due tothe device geometry, the capacitance that has the strongest dependence on theCNT displacement (in direction y) is the capacitance to the backgate, which wedefine as Cg(y). A charge q on the CNT is balanced by a charge −q on the gate.The Coulomb attraction between q and −q causes an electrostatic force that pullsthe CNT closer to the gate. This force can be calculated using F = −∇E, whereE is the free energy of the CNT as defined in Eq. (2.52):

Fy =1

2

∂Cg

∂yV 2g . (2.53)

Here Fy is the electrostatic force in the direction y, and Vg is the gate voltage.

In the presence of an oscillating gate voltage: Vg = V0 + Vg, Eq. (2.53) gives:

Fy =1

2

∂Cg

∂y

(V 20 + 2V0Vg

)for Vg � V0. (2.54)

The DC voltage V0 produces a static force on the CNT that can be used to tunethe tension (see also Sec. A.2.1). The AC voltage Vg produces on oscillatingforce that can set the CNT in motion when it is on resonance with a mechanicalmode. In reported experiments on suspended CNT devices capacitive drivingand tuning with a gate is standard. Oscillating voltages are either applied tothe gate directly [45–48], or by free-space coupling of the gate to a nearby RFantenna [43,49].

Transduction: mixingBecause CNTs are semiconductors, their conductance depends on doping. InFET-like structures as a suspended CNT, the charge on the CNT is the productof the CNT-gate capacitance and the gate voltage. The CNT capacitance dependson the position of the CNT. Since the capacitance is a function of position, theconductance changes when the CNT is displaced. This can be used to read-out

44

CNT vibrations. Suspended CNTs have resonant modes in the 10MHz . . . 10GHzrange. Low temperature amplification of small signals at these frequencies is nottrivial, and for this reason usually a mixing or rectification technique is used toconvert the signal to DC, which can be measured with low-bandwidth amplifiers.

Normally AC gate voltages have to be applied to drive the resonator to amplitudesthat are big enough to give a detectable signal. In many reported experimentsa mixing technique has been used to measure the mechanical motion [45–47]. Inthese experiments oscillating signals were applied to the gate and bias contactsin a controlled way, and could be separately tuned. Next we will derive a simpleexpression for the mixing current in such experiment.

When a CNT moves above the gate the induced charge q is (up to first order):

q = CgVg = C0Vg + V0Cg . (2.55)

Here the first term is the standard FET effect and is present at any frequency. Thesecond term is non-zero only on mechanical resonance, because the capacitancewill only change when a vibrational mode of the CNT is excited. The DC mixingcurrent is given by:

Im =⟨VsdG

⟩. (2.56)

Here Vsd is the AC bias voltage. Brackets denote time averaging and tildesdenote the time-varying parameters. The conductance can be (up to first order)expanded for small q in the following way:

G =∂G

∂qq =

∂G

∂q

(C0Vg +

∂C0

∂yyV0

). (2.57)

Which can be rewritten as:

G =∂G

∂Vg

(Vg +

V0

C0

∂C0

∂yy

). (2.58)

The mixing current Im is now found:

Im =∂G

∂Vg

(⟨VgV

⟩+

V0

C0

∂C0

∂y

⟨yV

⟩). (2.59)

The product of two time varying signals with ω1 and ω2, gives a signal containingspectral components at ω1+ω2 and ω1−ω2. A DC signal is generated only wheny or Vg is oscillating on resonance with the bias voltage V at frequency ω. Inthat case the mixing signal has a component at ωm = 0 and ωm = 2ω. The highfrequency component is filtered out and the DC component is detected, whichgives a signal:

Im,0 =1

2

∂G

∂Vg

(⟨VgV

⟩+

V0

C0

∂C0

∂y

⟨yV

⟩). (2.60)

In the following discussion we find it convenient to refer to the factor ∂G/∂Vg,as the transconductance1.

1Normally the transconductance is defined as g = ∂I/∂Vx, where I is the output currentdue to a voltage change ∂Vx somewhere in the circuit. Since ∂I = ∂VsdG, our definition oftransconductance is g/Vsd = ∂G/∂Vg. From now on when we talk about transconductance, weimply ∂G/∂Vg.


The amplitude of Im,0 also depends on the phase between the oscillating signals:

Im,0 = Idirect cos(ϕ1) + Imech (Re [yω] cos(ϕ2)− Im [yω] sin(ϕ2)) . (2.61)

Here Idirect and Imech follow from Eq. (2.60) and yω is the response function of theCNT resonator. We have defined the phase between y and V as ϕ1, and betweenVg and V as ϕ2. The phases ϕ1,2 are set by the electrostatic environment ofthe device and cannot be controlled in our experiments. The phase can changeas a function of frequency, which results in different peak shapes, ranging fromLorentzian lineshapes to Fano lineshapes [46].

In the absence of a bias voltage, mixing can still occur due to a mixing term with⟨Vgy

⟩. In App. B we present a detailed derivation of the change of conductance

of a vibrating CNT in the presence of only an oscillating gate voltage. There we

derive that for resonators with Q � 103 the term⟨Vgy

⟩is small, resulting in a

response that is Lorentzian in the absence of oscillating bias voltages. Accord-ingly, when Fano lineshapes are observed in resonators with Q � 103 this is a

strong indication of mixing originating from the term with⟨yV

⟩in Eq. (2.60).

We point out here that the mixing current is proportional to the transconduc-tance ∂G/∂Vg, and that the sign of the signal is independent of the DC voltageacross the device. In Fig. 2.20a we present how mixing can be distinguished fromrectification in a bias-dependence measurement.

In Fig. 2.20b we show a range of Fano-lineshapes, characterized by the parameterq. We have used the following Fano-lineshape equation to fit our data [50]:

A(ω) =(qΓ/2 + ω − ωr)

2

(ω − ωr)2+ (Γ/2)

2 , (2.62)

with the purpose of extracting the lineshape width Γ, the resonance frequencyω/(2π) and the lineshape size. We also use this fit to find the “zero” of ourresonator lineshapes. Here we define zero as the off-resonant voltage. Noticethat the Fano function in Eq. (2.62) goes at extremities to one rather than zero,in practice we fit to A(ω)− 1.

Transduction: rectificationIn App. B we discuss rectification and mixing in detail. Here we will present themost important features of rectification. Rectification of a time-varying signaloccurs in any component that has a non-linear IV -characteristic. A well knownrectifier is a diode. In CNT resonators the non-linear transconductance canbe used to rectify any signal that is equivalent to an oscillating gate voltage.Rectified current is given by:

Ir =⟨G⟩Vsd , (2.63)

where⟨G⟩is the time average of the non-linear conductance at a specific gate

voltage, and Vsd is the DC component of the voltage across the device.

We point out here that the rectification current is proportional to the first deriva-tive of the transconductance ∂2G/∂V 2

g , and that the sign of the signal changes

46

2.5. Vibrating suspended carbon nanotube Josephson junctions

q=1res=

q=1.8 q=3.2 q=5.6 q=103

res= /2

0

1

ampl

itude Q= /

Vsd=0

0

I y=G

VVsd>0Vsd<0

0

I y=G

V

rectification

mixing

(a)

(b)

Figure 2.20: Mixing vs. rectification as a function of bias. (a,top) Lorentzianlineshapes are typical for rectification. No signal is present at zero bias, and the signalchanges sign when the polarity is changed. (a,bottom) Asymmetric Fano lineshapesindicate mixing. Signal is present at zero bias, and the signal does not change sign whenthe polarity is changed. Iy is part of the current through the CNT that depends on themotion. (b) The asymmetry of Fano lineshapes is determined by the phase differencebetween the mixing signals on resonance. It is characterized by the parameter q. Herewe have plotted several (normalized) Fano lineshapes and indicated how the Q-factoris defined as a function of peak width.

with the polarity of the bias voltage Vsd. In Sec. B.1.3 we discuss how rectificationgives rise to Lorentzian lineshapes.

It should be noted that in the presence of microwave signals any CNT devicewith a gate and source-drain contacts will (to some extend) show mixing andrectification behavior. Because the gate is usually (asymmetrical) capacitivelycoupled to the source drain contacts, an oscillating voltage on the gate will resultin an oscillating voltage across the device as well. For these reasons it is difficult topredict whether mixing or rectification will dominate in our devices. Investigationof the gate voltage and bias dependence of the resonator signal can be used todetermine the dominant mechanism. In Sec. B.2 we present a numerical analysisof the gate dependence of time averaged conductance in the case of rectificationand mixing.

47


2.5.2 Capacitive coupling in a suspended CNT Josephsonjunction

In a suspended CNT Josephson junction in principle all the mixing and rectifi-cation signals that occur in a normal suspended CNT can also be present. Alsothe Coulomb force driving mechanism is present. In this section we discuss amechanism for additional driving of the nanotube motion, and additional mixingsignals that arise due to the Josephson effect.

Driving: Josephson forceThe Josephson energy depends on the charge on the nanotube, which in turndepends on the position of the CNT. This makes the Josephson energy a functionof nanotube displacement. A force will be exercised on the CNT that pulls thenanotube to a point where it has minimum Josephson energy. In Fig. 2.21a weshow how the total electrostatic energy is modified by the Josephson energy.

We have taken a simplified approximation, where we let the Josephson energyoscillate as a function of charge. The point of minimum energy can be tuned withthe gate voltage, such that it is on a point where dEJ/dq is large. In Fig. 2.21bwe have plotted a single oscillation of Josephson energy. In a CNT this wouldcorrespond to a peak in Isw(Vg). The Josephson energy can be minimized bydisplacement of the CNT towards lower energy. This results in a force that wecall Josephson force:

F J = −∇EJ . (2.64)

0

E

q(Vg)

EJ( )

q2/2Cg

-qVg

Esum

q(V0)

EJ(

)E

J(0)

(t)

q(t)

(b)(a)

Electrostatic energy Josephson energyFJ=- EJ

Figure 2.21: Josephson force. (a) Total electrostatic energy. (b) Josephson forcepulls the CNT to a point of minimum Josephson energy. The Josephson energy canoscillate in time when the phase is running. This results in an oscillating force actingon the CNT.

48

2.5. Vibrating suspended carbon nanotube Josephson junctions

In general the direction of the force will be such that the CNT is being pulledtowards the gate. Furthermore, in case of a constant voltage bias, the Josephsonforce will oscillate with ω1 = 2eV1/�. Motion will be driven when the Josephsonforce is on resonance with a mechanical mode. This implies that 2eV1/h ∼ 1GHz,for realistic CNTs with 1GHz mechanical modes. A voltage bias of V1 = 2μVcan be maintained across the junction without much trouble, for example byirradiating the junction with a 1GHz RF signal.

The most important limiting factor, and probably the reason why Josephson forcedriving will be very small, is that it is almost impossible to make an efficientenergy transfer between the Josephson junction and the CNT mechanics. This isdue to the large mismatch between the linewidth of the CNT mode (∼ 10 kHz)and the AC Josephson linewidth, that is determined by noise and can easilybe 1MHz (taking 2 nV voltage noise). We note (this was discussed in Sec. 2.3.3)that we do not have a complete understanding of the effect and amount of voltagenoise in our system, and that 1MHz is a conservative guess.

Under the assumption that the linewidth of a CNT mode and the AC Josephsoneffect are somehow matched, we have estimated that the Josephson force can bevery large and be the dominant driving force at low power (see App. C). TheJosephson force has a peculiar power dependence that is presented in Fig. 2.22.It is a smoking gun signature in which Josephson driving can be distinguishedfrom gate force driving.

Josephson force

F/F c

1

-0.5

0.5

0

eV1/ 1

0 5 10 15 20 25

Figure 2.22: Comparison of power dependence of gate force and Josephsonforce. The Josephson force has a Bessel function dependence on V1, and is maximumat low driving voltage. The gate force increases linearly with the driving voltage.

The power dependence of the Josephson force has a Bessel-function shape. InApp. C we have analyzed this thoroughly. Unfortunately we have to stress thatits relevance is likely to be minimal in our experiments due to the potentiallyvery large linewidth mismatch.

49


Transduction by mixing with Josephson currentThe Josephson energy of a CNT Josephson junction is a function of charge. Whenthe CNT moves above a gate, this changes the charge on the CNT and that inturn changes the Josephson energy. This results in mixing of motion with ACJosephson currents that on resonance can give rise to a time averaged DC signalIm, in the following way:

Im =1

2

∂Ic∂q

∂q

∂y〈y sin (ϕ)〉+ h.o. . (2.65)

Here Ic is the critical current, q is the charge on the CNT, y is the CNT motionand ϕ is the phase difference across the junction. Higher order terms (indicatedby h.o.) can also play a role. We can rewrite this equation in the following way:

Im =1

2

∂Ic∂Vg

V0

C0

∂C

∂y〈y sin (ϕ)〉+ h.o. . (2.66)

Josephson mixing or normal mixing?We will now compare the relative size of Josephson mixing and normal mixing.Using Eqs. (2.66) and (2.60) we make an order of magnitude estimation:

Im,J

Im,N=

(∂Ic∂Vg

sin ϕ

)/

(∂G

∂VgV

)≈ 10μV

V, (2.67)

where we have taken sin ϕ ≈ 1, and used typical values for our devices: ∂Isw/∂Vg =3nA/V (taking Ic = Isw) and ∂G/∂Vg = 0.4mS/V. In Fig. 5.9 we show that wecan measure the resonance signal at low power, before the first Bessel functionnode. In Fig. 2.15 we find that this regime corresponds to 0 < eV < 2.4hf whichis approximately 0 < V < 10μV at f = 1GHz. We estimate Im,J/Im,N ∼ 1 atthe smallest power at which we can still (practically) extract a mechanical signal.

In CNT Josephson junctions the theoretical critical current Ic is typically anorder of magnitude larger than the measured switching current Isw [22]. If thisalso applies here it might be that we underestimate Im,J/Im,N by one order ofmagnitude.

For simplification we have neglected effects of motion on the phase of the su-percurrent. When the nanotube moves above the gate the conductance changes,which can change the bias voltage across the device and that affects the phase.We have also neglected higher-order contributions.

With this we conclude our theoretical discussion on CNT mechanical vibrationsand superconductivity. The final part of this chapter will be dedicated to theeffect of a magnetic field on Josephson junctions. This discussion will be rele-vant for the experiments on graphene, but also for the experiments on vibratingnanotubes.

50

2.6. Josephson junctions in a magnetic field

2.6 Josephson junctions in a magnetic field

In this section we will discuss what happens to Josephson junctions when theyare placed in a magnetic field. First we will discuss Cooper pair-breaking, andsecondly Zeeman π-junctions. The section will end with a brief discussion onphase-sensitive measurement.

2.6.1 Magnetic field induced Cooper pair-breaking

When a Josephson junction is placed in a sufficiently large magnetic field, Cooperpairs will break up due to spin- or orbital depairing. In the paramagnetic limitthe Zeeman energy is larger than the Cooper pair binding energy Δ. In thatcase spins will align and Cooper pairs break. This pair-breaking field is set bythe Pauli-limit (Bc,Pauli = Δ/(gμB) = 3.52kBTc/(gμB)) and can be convenientlyexpressed as a function of the critical temperature Tc of the superconductor:

Bc,Pauli = 1.84Tc T , (2.68)

where we have taken g = 2.

In the orbital limit Cooper pairs are broken when the centripetal force due to theorbital motion of electrons around B separates the electrons in a Cooper pair bya distance larger than the coherence length (Cooper pair size) ξ0. This limit canbe expressed as Bc2 = Δ/ (ξ0evF), where Δ is the (proximity) gap, and vF is theFermi velocity. Substitution of Eq. (2.18) yields the expression:

Bc2 = φ0/2πξ20 , (2.69)

each orbit can accommodate one flux quantum φ0.

The orbital pair-breaking field can be increased by confinement of the orbital mo-tion. If for example the physical dimension (thickness) d of the Josephson junctionfacing the magnetic field is smaller than the coherence length: Bc2 = φ0/2πd

2.Due to their dimensions, graphene and CNT Josephson junctions are very suit-able for orbital confinement and the orbital pair-breaking field is expected to bevery large. Since CNTs are basically one-dimensional conductors where electronscan only move forward, backward, or around the CNT, orbital pair-breaking isexpected to occur only at magnetic fields exceeding B > 40T. In (flat) grapheneorbital pair-breaking is completely suppressed when the field is exactly in-plane.

When Bc2 < Bc,Pauli, the pair-breaking field is in principle set by the Pauli-limit.Spin-orbit interaction can counteract spin alignment, and in superconductingnanowires with strong spin-orbit interaction critical fields exceeding the Pauli-limit have been found [51].

Proximity induced gaps of Tc ∼ 0.8K are common in graphene [52] and CNT [18]junctions with Al electrodes. Therefore we expect that such junctions should beable to carry supercurrent at in-plane magnetic fields as high as 1.5T.

51


2.6.2 Fraunhofer patterns in 1D ballistic junctions

Characteristic Fraunhofer patterns of Isw(B⊥) are typically observed in Josephsonjunctions. Fraunhofer patterns reflect the current distribution in the junction.Oscillations are periodic with Φ/Φ0 [34]. Switching current decay in a Fraunhoferpattern is the most common way supercurrent is reduced in a magnetic field. InCNT Josephson junctions the area is so small that Φ/Φ0 = 1 is reached atB = 4 . . . 7T, in typical junctions with L = 200 . . . 400 nm and r = 1.4 nm.

In ballistic junctions where L/W → ∞, the Fraunhofer pattern period can bedoubled to Φ/Φ0 = 2 [53, 54]. This implies that in our CNT devices the firstnode in the Fraunhofer pattern is expected at B ∼ 10T, and therefore reductionof the switching current by Fraunhofer interference is probably not important atall.

2.6.3 Zeeman π-junctions

A π-junction is a Josephson junction where the ground state is at a phase ofϕ = π instead of ϕ = 0. In this section we will discuss π-junctions formed bymagnetic field. First we will give a short motivation for research on π-junctions.

Why π-junctions?In superconducting logic circuits, Josephson junctions are used as switching el-ements in a way transistors are used in semiconducting logic circuits [55]. Thistechnology has important advantages over traditional semiconductor technology.Power consumption is much reduced because of the superconducting nature of theelements in combination with relatively low operating current/voltages. Joseph-son junctions can also be switched very fast, which makes superconductive logicpotentially much faster than semiconductor logic. It is in this technology thatπ-junctions have specific applications, in particular they allow for a miniaturiza-tion of the circuits. In the field of superconducting quantum circuits π-junctionsalso have potential applications [56].

By tuning the device parameters (i.e. temperature, magnetic field) a 0− π tran-sition can be induced. There it is possible to study higher order components (likesin (2ϕ)) of the current phase relation that are usually masked by the first-ordersin (ϕ) component. In a Zeeman π-junction it is possible to tune through thetransition by applying a magnetic field. This has advantages above existing π-junctions in S/F/S devices, where the tuning is done by varying the thickness ofthe F layer, or temperature change [57–59].

Physics of Zeeman π-junctionsWe consider two types of junctions, diffusive and ballistic. We will start with themodel that holds for general diffusive junctions with many ABS. At the end wediscuss a model that holds for ballistic junctions with few ABS. The first modelis applicable to our experiments on graphene, and the second model is probablycloser to our experiments on CNTs.

52


When a Josephson junction is placed in a magnetic field, spin-up/down electronsare split in energy by the Zeeman energy. Electrons with opposing spin in Cooperpairs compensate the Zeeman energy by adjusting their kinetic energy. Theyacquire a net center of mass momentum −hΔk for spin-up/down or hΔk for spin-down/up. This can be written as: |↑↓〉 = eiΔkxχ↑↓(x) and |↑↓〉 = e−iΔkxχ↓↑(x).

Because Cooper pairs are spin singlets, the order parameter isΨ = 1/

√2(|↑↓〉− |↑↓〉), and it oscillates as a function of x in the following way [60]:

Ψ(x) = Ψ(0)e−x/ξd1/2(e−iΔkx + eiΔkx

)= Ψ(0)e−x/ξd cos (Δkx) , (2.70)

where ξd accounts for the decay of Cooper pairs in the non-superconductingmedium. In Fig. 2.23 we give a simplified graphical explanation of Zeeman π-junctions.

Emin at =0 Emin at =

E

-

E

-

x

g B

x x

-junction0-junction Zeeman 0- junction

g B+ k k

E

E

EF

- k k

(a)

(c)

(b)

(d)

Figure 2.23: Zeeman π-junction. (a) In a normal 0-junction the ground stateorder parameter Ψ is an even function across the junction. In a π-junction the groundstate order parameter Ψ is an odd function across the junction. (b) A Zeeman 0-π-junction can be made by employing the oscillating order parameter in a B field. (c)In a normal 0-junction the Josephson energy is minimum at ϕ = 0. In a π-junctionthe Josephson energy is minimum at ϕ = π. Stars indicate phase at ground stateenergy. (d) Electron (quasiparticle) subbands are split by the Zeeman energy in amagnetic field. The Zeeman energy is compensated by kinetic energy, spin down/upelectron pairs acquire a momentum Δk, spin up/down electron pairs a momentum −Δk.Electron pairs are indicated by black dots in the diagram.

By changing the magnetic field, the period of the wavefunction oscillation changes.The order parameter in the N part is determined by the sum of the wavefunctioncoming from the left and the wavefunction coming from the right. We have plot-ted symmetric (ϕ = 0, dashed) and antisymmetric (ϕ = π, dotted) combinations.The sum is as follows:

Ψ(x) = Ψ(0, A)e−x/ξd cos (Δkx)+Ψ(0, B)eiϕe(x−L)/ξd cos (Δk(x− L)) . (2.71)

53


When L � ξd, Ψ(0, A) = Ψ(0, B) = Ψ0. By substitution of this equation intothe expression for the quantum mechanical current: J ∼ Ψ∇Ψ∗ − Ψ∗∇Ψ, andtaking x = L, the following current-phase relation can be obtained [61]:

I(ΔkL) ∝ sin (ϕ) [cos (ΔkL) + sin (ΔkL)] e−L/ξd . (2.72)

We have plotted I(ΔkL) for ϕ = π/2 (critical current) in Fig. 2.24.

We observe that |I(ΔkL)| exhibits nodes. In our experiments ΔkL can bechanged by a magnetic field, as was discussed above. By calculation of theGinzburg-Landau free energy of the order parameter in Eq. (2.71), it can beshown that the ground state changes from symmetric to antisymmetric at a crit-ical field B0−π [61]. This point corresponds to the first node in Fig. 2.24. Atsuccessive nodes the order parameter symmetry changes again. In the π-state

0 2 4 60

0.5

1

kL

I c/I c

,0

0 0

Figure 2.24: Dependence of the critical current on ΔkL. The dashed lineindicates the absolute value of the critical current.

the ground state of the junction is reached at ϕ = π, changing the current-phaserelation I(ϕ) → I(ϕ + π). In the case of a sinusoidal current-phase relationIc sin(ϕ) → Ic sin(ϕ+ π) = −Ic sin(ϕ). For this reason people sometimes refer tosupercurrent-reversal in π-junctions [30, 62–64].

In estimating B0−π we have to distinguish between theory for diffusive junctionsand theory for ballistic junctions. Experimentally π-junctions by exchange inter-action have been reported in diffusive S/F/S systems [57,58,65,66], and they havebeen pursued by an applied in-plane field (Zeeman π-junction) in diffusive S/N/Ssystems [67]. A recent review can be found in Ref. [68]. For completeness we men-tion here that recent experiments on Josephson junctions with InSb-nanowirespursued in our group also indicate Zeeman-π transitions.

Estimation of B0−π for diffusive junctionsWithout going into details we will give here a theoretical prediction of B0−π fordiffusive junctions [69, 70]:

gμBB0−π = 16ETh = 16�D/L2 . (2.73)

54


Typical values are: B0−π ∼ 1 . . . 2T for graphene (with ETh = 10μeV [71]), andB0−π ∼ 1 . . . 2T for InAs nanowires (with ETh = 100μeV and g ∼ 10 [24,72,73]).We notice here that graphene is better suitable then InAs NWs, because orbitaldepairing will be suppressed as electrons are confined to the plane of graphene.In fact InAs NWs are not suitable because orbital depairing will probably preventobservation of supercurrents at B = 1 . . . 2T [67].

After we had stopped our experiments on graphene, three experiments on simi-lar graphene Josephson junctions as we have made, reported a Thouless energyranging from ETh = 80μeV to ETh = 240μeV [74–76]. Following Eq. (2.73)such high Thouless energy would shift B0−π > 10T, which makes it inconceiv-able to observe a 0− π transition in such devices. We notice that decreasing themean free path of charge carriers in graphene, for example by making graphenemore “dirty”, will lower the diffusion constant and decrease the Thouless energy,making it easier to observe the 0− π transition.

Order of magnitude estimation of B0−π for ballistic junctionsIn the case of a ballistic CNT junction with only few ABS, we can get an ap-proximation of B0−π by considering the effect of Zeeman splitting on ABS. Weconsider a single spin- and valley-degenerate ABS in a CNT. The ABS is four-fold degenerate at B = 0, and has an E(ϕ) relation according to Eq. (2.21). InFig. 2.25 we present our simple model of a ballistic π-junction. At B = 0 we plotthe usual ABS spectrum in panel (a), ground state is at ϕ = 0. With a magneticfield the ABS are split by the Zeeman energy (panel (b)).

When gμBB ∼ Δ (panel(c)), the ABS cross. We have drawn anticrossings,hybridization could occur due to for example spin-orbit interaction. In this regimethe current phase relation has an E(2ϕ) character.

At gμBB ∼ 2Δ (panel(d)) the ABS have crossed and form an inverted dispersionrelation compared to B = 0. The Josephson energy is now Eπ(ϕ) = −EB=0(ϕ) =EB=0(ϕ+π), the groundstate is at ϕ = π, indicating the presence of a π-junction.

0 2

0EA

BS

0 2

0

0 2

0E

E

g B

0 2

0 so

B=00-junction

B 0>~ g B~sin(2 )-junction

g B~2-junction

(b) (c) (d)(a)

+ - - +

Figure 2.25: Ballistic Zeeman π-junction. (a) Energy of ABS as a function ofphase, at B = 0. Ground state energy of the junction is at ϕ = 0. ABS are four-folddegenerate. The ± symbols indicate dE/dϕ. (b) A magnetic field splits the spin-degenerate ABS by gμBB. (c) At gμBB ∼ Δ the junction has a sin(2ϕ) nature. (d)At gμBB ≥ 2Δ the ground state energy is at ϕ = π.

55


We define the 0 − π transition field by B0−π = 2Δ/(gμB). For our junctionsΔ ∼ 75μeV (see Fig. 5.7), which yields B0−π = 1.3T. This gives only an orderof magnitude estimation, B0−π ∼ 1T.

We can think of two mechanisms that influence the magnetic field acting on ABS:Field focussing and screening. These mechanism are complementary and due tothe Meissner effect. In type-I superconductors the magnetic field is expelled com-pletely, and in type-II superconductors the field is partly expelled [34]. Whereasbulk rhenium is known to be a very good type-I superconductor (up to 99% fluxexpulsion has been reported [77, 78]), we can only guess the nature of our ownsputtered thin films. Impurities added to the film during sputtering can changeits properties, and possibly change the film to have type-II character.

Flux expulsion leads to a phenomenon called field focussing that enhances the fluxdensity between two closely spaced superconductors when placed in a magneticfield, and can be relevant in π-junctions [59]. In our experiment it would implythat B0−π < 2Δ/(gμB).

We also have to consider that ABS extend on the scale of the coherence length,that we estimate at 3.5μm in our devices (see Eq. (2.18)). Since our CNTs areabout 300 nm long, the ABS exist mainly in the SC contacts, where the magneticfield is (partly) expelled. This would result in B0−π > 2Δ/(gμB).

Critical current as a function of field for ballistic junctionsWe expect that in the case sin(2ϕ) components have a considerable contribu-tion to the supercurrent, the switching current vs. field dependence can deviatestrongly from that shown in Fig. 2.24. In Fig. 2.26 we sketch a possible scenariothat can account for a foot structure rather than a cusp in Isw(B).

0 2 4 60

0.5

1

kL

I c/I c

,0

sin(2 )sin( )

0 2 4 60

0.5

1

kL

I c/I c

,0

(b)(a)

Figure 2.26: Foot structure in Ic(B). (a) Critical current as a function of ΔkL ∝B. Grey curve is for the component Ic1 sin(ϕ) and black curves are two scenarios forIc2 sin(ϕ). (b) The total critical current has a foot-like structure.

The total supercurrent is set by the sum of sin(ϕ) and sin(2ϕ) components, whichhave a different dependence on ΔkL ∝ B. That implies that at B0−π the su-percurrent is not zero, but carried by the sin(2ϕ) component that is still sizableat this point. The presence of sizable sin(2ϕ) components across the whole fieldrange can cause fractional Shapiro steps, as will be discussed in Fig. 2.27. Theo-retically foot-structure field dependence of critical current has been predicted inballistic SFS junctions by Chtchelkatchev et al. [79].

56


Half integer Shapiro steps at 0− π transitionOne characteristic feature of the 0−π transition is suppression of the sin(ϕ) termof the Josephson current phase relation, and the dominance of higher order termslike sin(2ϕ). This is graphically shown in Fig. 2.25. The current-phase relationof such residual supercurrent can be studied in Shapiro step experiments. Thesin(2ϕ) term causes fractional Shapiro steps at Vn = (n+ 1/2)�ω1/(2e) [66].

In Fig. 2.27 we give a graphical explanation for fractional Shapiro steps.

0 5IC 0 -IC IC 0 -IC

V1

0

8

t (2 J)-1

(a) (b) (c)

V1/2

Figure 2.27: Fractional Shapiro steps due to sin(2ϕ). (a) Ic1 sin(ϕ). (b)Ic2 sin(2ϕ), with Ic2 > Ic1. (c) Where phase locking to sin(ϕ) results in Shapiro steps atV1, phase locking to sin(2ϕ) is already possible at V1/2, resulting in fractional Shapirosteps.

2.6.4 Phase-sensitive measurement

The phase of the supercurrent can be addressed directly in a phase-sensitivemeasurement. The most straightforward method for such measurements includeincorporation of the device under study in a SQUID. A SQUID consists out of twoJosephson junctions connected in a loop [34]. Magnetic flux φ penetrating theSQUID area causes a phase being picked up by electron-hole pairs going roundthe loop. Interference leads to a characteristic dependence of Ic(φ). A shift of πin one of the two junctions is then indicated by a shift of Ic(φ) → Ic(φ+ φ0/2).A direct phase-sensitive measurement of the current-phase relation of a singlejunction can be done by incorporating the junction in a SQUID, or by couplingthe junction inductively to an external SQUID [58,74,80].

57

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[32] K. K. Likharev, Superconducting weak links, Rev. Mod. Phys. 51(1), 101–159 (Jan 1979).

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[36] B. D. Josephson, Possible new effects in superconductive tunnelling, PhysicsLetters 1, 251–253 (July 1962).

[37] S. Shapiro, Josephson Currents in Superconducting Tunneling: The Effectof Microwaves and Other Observations, Phys. Rev. Lett. 11(2), 80–82 (Jul1963).

[38] D. E. McCumber, Effect of ac Impedance on dc Voltage-Current Character-istics of Superconductor Weak-Link Junctions, Journal of Applied Physics39(7), 3113–3118 (1968).

[39] C. Wylie, Advanced Engineering Mathematics, McGraw-Hill Book Com-pany, 1960.

[40] A. Dahm, A. Denenstein, D. Langenberg, W. Parker, D. Rogovin, andD. Scalapino, Linewidth of the radiation emitted by a Josephson junction,Physical Review Letters 22(26), 1416–1420 (1969).

[41] A. Silver, J. Zimmerman, and R. Kamper, Contribution of thermal noise tothe line-width of Josephson radiation from superconducting point contacts,Applied Physics Letters 11(6), 209–211 (1967).

[42] S. Sapmaz, Y. M. Blanter, L. Gurevich, and H. S. J. van der Zant, Carbonnanotubes as nanoelectromechanical systems, Phys. Rev. B 67(23), 235414(Jun 2003).


[44] L. Landau and E. Lifshitz, Theory of Elasticity, Pergamon Press, 1959.


[46] B. Witkamp, M. Poot, and H. Van Der Zant, Bending-mode vibration of asuspended nanotube resonator, Nano letters 6(12), 2904–2908 (2006).

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[53] J. Heida, B. Van Wees, T. Klapwijk, and G. Borghs, Nonlocal supercurrentin mesoscopic Josephson junctions, Physical Review B 57(10), 5618–5621(1998).

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[55] K. Likharev, Rapid single-flux-quantum logic, Nato Asi Series E AppliedSciences 251, 423–423 (1993).

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[57] T. Kontos, M. Aprili, J. Lesueur, F. Genet, B. Stephanidis, and R. Boursier,Josephson junction through a thin ferromagnetic layer: negative coupling,Physical review letters 89(13), 137007 (2002).

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[65] V. Ryazanov, V. Oboznov, A. Rusanov, A. Veretennikov, A. Golubov, andJ. Aarts, Coupling of Two Superconductors through a Ferromagnet: Evi-dence for a π Junction, Physical Review Letters 86(11), 2427 (2001).

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63

Chapter 3

Device fabrication

In this chapter we will discuss the fabrication process of graphene and CNTdevices. They are very different, because in the case of graphene, the graphene isplaced on a wafer first and the contacts come on top. In the case of CNT devices,the contacts are fabricated first and the CNT is placed on top. In Sec. 3.1 wewill briefly discuss electron beam lithography. In Sec. 3.2 we will discuss thefabrication of graphene Josephson junctions, and in Sec. 3.3 the fabrication ofclean suspended CNT Josephson junctions. We will conclude with Sec. 3.4, wherewe introduce the preselection procedure that we have followed to find good CNTdevices. We have included all detailed process flows in App. E.

3.1 Electron beam lithography

We have made devices with graphene and carbon nanotubes. Contacts and gateswere made that provide electrical access and control on graphene and nanotubes.This was done with lithographic techniques. Because the required dimensions areusually below the wavelength of visible light, electron-beam lithography (EBL)is used rather than optical lithography. With a commercial e-beam writer suchas the one used in this work, structures with dimensions on the order of ∼ 10 nmcan be made. Alignment of different e-beam writing steps is possible with anaccuracy on the order of ∼ 50 nm. This can be important when two structureshave to be placed closely to each other.

3.2 Fabrication of graphene Josephson junctions

The fabrication of graphene Josephson junctions is a three step process. In thefirst step, the wafer is cleaned and a marker pattern is fabricated on the surface.This is the first EBL step. In the second step, the SiO2 substrate is cleanedagain and graphene is deposited. The third step is the e-beam step during whichsuperconducting contacts are patterned on the graphene surface. In this sectionwe will discuss the basic process steps.

65

3. Device fabrication

3.2.1 Wafer cleaning

Special attention has been made to clean the wafer before graphene deposition.The reason for this is that the potential landscape in graphene is easily affectedby residual charge in its vicinity (on the substrate). This can lower graphene mo-bility, and have an effect on the conductance and supercurrent through graphene.We have followed the industry standard SiO2 wafer cleaning procedure, RCA1and RCA2 [1]. This method has been known to give satisfying results for graphenedevices made at UBC [2]. RCA1 cleans the substrate from insoluble organiccontaminants. RCA2 cleans the substrate from ionic and heavy metal atomiccontaminants.

3.2.2 Graphene deposition

Graphene is very easy to make. Needed are a piece of glass, a pencil and a piece ofsticky tape. Since its discovery in 2004 the graphene deposition process has beenrefined and reproduced by countless groups around the world. Each group has itsown tips and tricks. In essence most recipes do not really differ from the originalone. In Fig. 3.1 an example is given of graphene preparation. Differences can layin choice of substrate, pre-deposition substrate treatment, type of graphite, typeof sticky tape, number of peelings, duration and amount of pressure applied tothe tape/graphene/substrate and post-deposition treatment.

After graphene deposition the surface is usually covered with a bit of grapheneand a lot of graphite. Single layer graphene can be distinguished from multi-layer graphene, simply by comparing the contrast in optical microscope images(see supplementary information in Ref. [4]). Also Raman spectroscopy can beused for this purpose [5]. Graphene is found with a microscope and logged withan image that is later to be imported in a CAD program.

3.2.3 Thin superconducting contacts on graphene

We have used three different type of superconducting contacts on graphene. Theproperties of these contacts will be presented in Sec. 4.4. Because the Ti/V/Titrilayer contacts gave the best results, we will (only) report on those here.

In Fig. 3.3 the fabrication process for the contacts is shown. We have used atrilayer contact of Ti/V/Ti. This structure has been specifically developed forthe purpose of making very thin superconducting contacts to graphene. Thetrilayer has two advantages. Firstly, it can be made as thin as ∼ 15 nm while stillelectrically continuous and superconducting. Secondly, the trilayer can be madeby e-beam evaporation rather than sputter deposition. With sputter depositiongraphene can be etched away by the plasma that facilitates sputtering.

The recipe for the trilayer structure has been taken from Tedrow and Meservey [6],we found that it was very easy to reproduce their results in our AJA system. Thestructure is Ti/V/Ti with 3.5/10/3.5 nm thickness. Titanium acts as a cappinglayer to vanadium and prevents it from oxidation.

66

3.2. Fabrication of graphene Josephson junctions

Figure 3.1: Mechanical exfoliation of graphene. (a) Optical microscope imageof a natural graphite grain of roughly 1.5 × 1.5mm2. (b) Repeated peeling of naturalgraphite with Nitto tape. (c) Inspection with optical microscope to find graphene. (d)Optical microscope image of a graphene flake. A marker used to locate graphene isalso visible in the upper right corner of the picture. The black dashed line indicatethe region where the AFM image in (e) is taken. (e) Atomic force microscope (AFM)image of part of the graphene flake in (d). Figure adapted from Ref. [3].

150nm ZEP Ti/V/Ti trilayer

ZEP on graphene

e-beam lithography

metal deposition

anisol lift off

graphene

500 m p++ Si 285nm SiO2

Figure 3.2: Process flow for contact deposition on graphene. Graphene is spincoated with e-beam resist. With EBL the contact area is defined. By physical vapordeposition metal contacts are deposited on graphene. After lift off the device is ready.

Vanadium has a relatively large melting point (1910 �C) and high power is nec-essary for e-beam evaporation. A high deposition rate is favorable to reducethe amount of trapped impurities due to background pressure in the evaporationchamber. We have used ZEP520A instead of the more common PMMA. ZEPhas a (∼ 2.5) times lower etchrate compared to PMMA, and for this reason itwas picked for an earlier process that involved niobium sputtering. We have usedthis process for the Ti/V/Ti structures as well, because it gave good results.

67


ZEP has more advantages, it requires a smaller dose than PMMA (writing pat-terns goes an order of magnitude faster), while maintaining a very high resolution.A disadvantage is that it has more adhesion problems than PMMA. We foundthat a single layer of ZEP with ∼ 150 nm thickness facilitated a good lift-off forthe trilayer films on graphene. Typical junction W × L was W = 200 . . . 300 nmand L = 5μm.

250 m10 m

X

(a) (b) (c) (d)

Figure 3.3: Contact fabrication on graphene. (a) Optical microscope image(100×) of a graphene flake on the substrate. We have indicated the outline of thegraphene flake with a black line. Markers are visible left of the graphene flake. Scalebar is 10μm. (b) CAD design of contacts. The red area will be exposed by e-beam.The markers are used for alignment. (c) Contacts on graphene after EBL. (d) Zoomout shows larger contacts moving out to the bonding pads. Scale bar is 250μm.

3.3 Fabrication of suspended carbon nanotubeJosephson junctions

To get high electronic and mechanical quality devices, the CNT has to be clean(see Sec. 1.5). In the clean fabrication method a trench is made first, and thenanotube is grown across. It is not known beforehand where the good device willbe, because the direction and number of grown nanotubes is random and thereis no control on the bandgap of the nanotubes. In Fig. 3.4 we show a schematicpicture of a typical device.

catalyst superconductor dielectric

gate

CNT

Figure 3.4: Schematic of a CNT device. The suspended CNT device consists outof a CNT that was grown from a catalyst particle and collapsed across a trench etchedin rhenium on SiO2.

We have found that the fabrication process has a very low yield (∼ 1%), of findingmetallic nanotubes. In a similar fabrication process reported in 2005, also a yieldof ∼ 1 . . . 2% metallic nanotubes was found [7]. To get ∼ 10 good devices we havemade 1200 trenches on a 19× 19mm2 wafer. In Fig. 3.5 the CAD design of our

68

3.3. Fabrication of suspended carbon nanotube Josephson junctions

devices is shown. A good device consists out of a single, clean, small band gapnanotube across a trench.

(a) (b) (c)

450 m 25 m 3mm

Figure 3.5: CAD design for suspended CNT devices. (a) One cell is 3× 3mm2

and has 48 contacts to 24 devices. The black part will be etched away, white is remainingmetal. (b) Zoom in on a cell, showing 24 devices. (c) Zoom in on two devices. Thecatalyst is deposited on a 1× 1μm2 square, marked in grey.

Preselection by scanning electron microscope (SEM) is not favorable because itcan cause deterioration of the nanotube properties by inducing structural changes,or carbon deposition on the CNT. This was reported elsewhere (see Refs. [8, 9]),and also found by us (see Fig. 5.4). Preselection by AFM is also not favorablebecause it is time consuming and difficult on suspended nanotubes. Unfortu-nately it is usually only after a cool down in a dilution fridge that the quality of adevice can be truly judged, by inspection of the electrical and mechanical charac-teristics. A good device has a clear gate dependence of the conductance and canhave mechanical modes with very low damping. We have followed an electricalpreselection process to select devices that are suitable for the experiment. Thisprocess will be discussed in Sec. 3.4. In this section the fabrication steps will beintroduced.

3.3.1 Rhenium as contact for clean CNT fabrication

It is important that the contact metal is compatible with nanotube growth con-ditions. The art of clean nanotube fabrication can be summarized as findingthe right wafers and metals to make devices. To make a Josephson junction therequirements are stringent:

� The contact must be a superconductor,

� transmission for electrons or holes at the SC/CNT interface should be high,

� contacts should not melt or react badly with methane or catalyst duringgrowth.

Rhenium combines all the above requirements. Bulk rhenium is a superconductorwith Tc = 1.7K. In thin films Tc = 1.4 . . . 2.8K is found [10]. At atmospheric

69


pressure it has (after tungsten) the highest known melting point of all elements at3185 �C. Our tests on thin (∼ 20 nm) rhenium films have shown that they do notconsiderably change (structurally or electrically) after exposure to methane in thenanotube furnace (see App. D). The work function of rhenium is with 4.7 eV closeto the work function of SW-CNTs (single-wall CNTs), that is experimentallyfound to be φCNT = 4.9 ± 0.1 eV [11]. This is favorable for the formation oftransparent interfaces.

Because surface oxide is potentially a major obstacle to achieve transparent in-terfaces, we searched for the properties of rhenium oxides. Five oxides of rheniumare known:

Dirhenium heptaoxide Re2O7, forms when the metal is heated in air or oxy-gen above 150 �C. The (theoretically calculated) melting point is 300 �C,but the oxide sublimes [12].

Rhenium trioxide ReO3, decomposes in vacuum above 400 �C to ReO2 andRe2O7. It can be formed by heating Re2O7 in CO, Re or ReO2. Thisunusual oxide is metallic [13, 14].

Rhenium dioxide ReO2, can be formed from Re2O7 and H2 at 300 �C and fromRe2O7 and Re at 600 �C. It decomposes to Re above 500 �C [12,15].

Dirhenium pentaoxide Re2O5, thermally stable up to 200 �C [13].

Dirhenium trioxide Re2O3, decomposes to Re and ReO2 at 500 �C [13].

A naive inspection of this list leaves us optimistic, because all oxides either de-compose or change to other oxides at some point during a temperature sweepfrom RT to 900 �C and back. We can only speculate on what exactly will happenin the nanotube furnace. Considering the rhenium oxide properties it is likelythat the surface will change during growth and that part of the nanotube getsexposed to a conducting surface of rhenium.

Technical details of depositionWe have used 20 nm films of sputter deposited rhenium on the silicon oxide surfaceof a p-doped silicon wafer. The film is sputtered with a rate of ∼ 40 nm/min in2μbar of argon at a power of 500W. The background pressure of the vacuumsystem is on the order of 2× 10−8 mbar.

3.3.2 Dry-etching of trench and contacts

In one etching step, bonding pads, leads and contacts to the nanotubes are de-fined. We have used a quadruple etch mask. The first layer (on rhenium) is athin layer of PMMA, that facilitates acetone lift-off of the etch mask. In thisway we do not have to use nitric acid to remove photoresist, which we found todamage the film. The second layer is a layer of hard baked photoresist. This layerserves as an etch mask, mainly to silicon oxide. The third layer is a thin film oftungsten, that is used as an etch mask to hard baked photoresist, the PMMAlayer and partly rhenium. During the rhenium etch this film will be etched away.The top layer is a thin film of PMMA, that is used as an etch mask to tungsten.

70


The pattern is written in the top layer and after development it is transferredfrom one layer to the next. The process flow is depicted in Fig. 3.6.

Re on SiO2 etch mask e-beam

lithography SF6 etch W

O2 etch S1813 + PMMA

SF6 etch Re acetone lift off

CHF3 etch SiO2

500 m p++ Si 285nm SiO2 20nm Re

100nm PMMA 400nm S1805

7nm W 200nm PMMA

Figure 3.6: Trench fabrication. Process flow for etching a trench in Re on SiO2.A quadruple layer etchmask is used to allow for a ∼ 200 nm trench. The first PMMAlayer on Re allows for an easy lift-off of the etchmask in acetone.

Different process gasses are used to etch tungsten, photoresist, rhenium and sili-con oxide. This makes it a lengthy process step which can take about four hoursto complete. The etching process is crucial to the fabrication and has been specif-ically designed by us with the purpose of fabricating trenches on SiO2-rheniumwafers. We show SEM pictures of typical trenches in Fig. 3.7. With this recipe150 nm deep trenches can be etched with a width of approximately 200 nm, at ayield of 95%.

200nm trench 200nm 250nm 7 m

250nm trench

150nm 3.5 m

(a) (b) (c)

(d)

(e)

Figure 3.7: SEM micrographs of trench before CNT growth. (a) Angle SEMmicrograph of a 200 nm trench. (b) 250 nm trench. (c,d) Top view SEM micrographof 200 nm and 250 nm trenches. (e) SEM image of two trenches with catalyst positionmarked by white square. The distance between two trenches is 7μm.

71


3.3.3 Catalyst deposition

Catalyst is patterned close to the trenches by EBL and lift-off. The catalystdeposition and lift-off process has to be done with care, to prevent catalyst inthe lift-off solvent to land back on the surface of the chip when it is taken outof the beaker. Nanotubes will also grow from catalyst that lands randomly onthe chip surface and these tubes can in turn short contacts to potentially gooddevices. The catalyst used is a solution containing aluminum oxide, iron nitrideand molybdenum oxide.

400nm PMMA catalyst film

PMMA on Re e-beam

lithography catalyst

deposition acetone lift off

500 m p++ Si 285nm SiO2 20nm Re

Figure 3.8: Process flow for catalyst deposition on rhenium. PMMA is spincoated on Re. After e-beam lithography, a catalyst solution is gently dripped on thesurface. The last step is catalyst lift off.

3.3.4 CNT growth

A nanotube CVD recipe has been used that is known to give ∼ 10μm long,predominantly single wall CNTs. Growth takes place in a flow of methane at atemperature of 900 �C. To reduce contamination by residue on the inside of thequartz growth tube, it is to be cleaned before growth. In Fig. 3.9 the processflow for CNT growth is given.

growth at 900°C catalyst on Re device ready

Figure 3.9: Process flow for CNT growth. CNT growth is the final fabricationstep. The chip is put in a quartz tube furnace and brought in a flow of methane at900 �C. These conditions are maintained for ∼ 10 minutes, after which the growth isstopped.

Nanotubes grow in random directions from the catalyst particles. The SEMmicrograph in Fig. 3.10 shows a beautiful example of a nanotube grown from acatalyst particle.

When the chip is removed from the growth chamber the devices are ready. Thefinal stage before cooling down in the dilution fridge is selecting good devices.

72


Figure 3.10: SEM micrograph of nanotubes growing from catalyst. CNTsgrow in random directions from the catalyst. Catalyst is patterned near the trenchand by chance a CNT can fall across it. This SEM image shows a ∼ 10μm long CNTsticking out of the catalyst square.

73


3.4 Selecting good nanotube devices

To select good devices we have followed the following preselection procedure.

3.4.1 Preselection in room temperature probestation

RT probestation: Resistance measurementAll trenches are probed with 10mV DC bias and grounded backgate. The re-sistance is measured at each trench. For Josephson junctions, low resistancedevices (� 50 kΩ) are selected. A low resistance implies a large supercurrent(see Sec. 2.2.4). We notice that this selection method is biased towards selectingtrenches that have more than one nanotube across the trench, which is unfa-vorable. The purpose of a later selection step is to find out if there is just onenanotube across the trench.

RT probestation: Gate tracesBy taking a room temperature gate trace it is possible to see if a nanotube has alarge or a small bandgap. For rhenium contacts we find that for small bandgapnanotubes the conductance is high on the electron as well as on the hole dopedregion. Here it should also be mentioned that the room temperature gate tracecan change after a thermal cycle, which can make it very hard to preselect a gooddevice. In Fig. 3.11 we give an overview of the results of this characterizationstep.

3.4.2 Preselection in low temperature dipstick

The RT gate traces already give an indication on the quality of a device. By LTcharacterization information is gained on: The bandgap, LT conductance, barriertransparency and the number of nanotubes. Our LT characterization experimentsare done in a variable temperature insert dipstick system at a temperature of∼ 4K.

A good way of identifying a single nanotube is by looking for four-fold periodicshell filling [16]. This characterization step is useful to find good devices forexperiments in the dilution fridge. In Fig. 3.12 three typical LT gate traces ofgood devices are shown.

It can happen that a large bandgap and a small bandgap nanotube are bothcrossing one trench. In that case it could be that for low gate voltages four-foldshell filling is observed, but that for larger gate voltages the pattern is less regularbecause the large gap CNT starts to conduct as well. A small gap nanotube canalso be shunted by a very large bandgap nanotube. In that case the transportbarriers can be so high that the current through that CNT is below the detectionlimit. In such a case the small bandgap nanotube can actually work as a chargesensor to the large bandgap CNT, and the presence of another tube can still bedetected (see Ref. [17] and Ch. 6). In Fig. 3.13 we show a SEM picture of a devicewhere two CNTs are closely together across a trench.

74

3.4. Selecting good nanotube devices

0 1 2 3 4 50

10

20

30RC21BN=198 bin=2.5k

R (h/e2)

# de

vice

s

0 1 2 3 4 5R (h/e2)

RC21CN=301 bin=2.5k

6-6 00

0.4

0.8

Vg

(V)

I(A

)

RC21CV=10mV

-6 0 6V

g(V)

RC21BV=10mV

(a) (b)

(c) (d)

Different colors are different devices

Figure 3.11: Room temperature characterization of CNT devices. (a,b)Histogram of device resistance at Vg = −6V. The grey area denotes the 5 . . . 12.5 kΩresistance range. Devices are on two 3 × 3mm2 chips with 25 cells, 1200 trenches intotal. (c,d) Room temperature gate traces of 13 devices that were selected for LTcharacterization (series resistance is 4.2 kΩ).

The presence of a second nanotube can also be observed by bias spectroscopy.This is shown in Fig. 3.14. The device A of Fig. 3.14 is an example of onepromising device that has been selected for cool down in the dilution fridge.

75


0

2

4

6

I(nA

)

T~2.5KVBIAS=~100 V 1

5 9 13

1

5

9131721

0

0.4

0.8

I(nA

)

T~4K

1

2 3 45 8

12 16 20 24

-6 -4 -2 0 2 4 60

0.2

0.4

Vg(V)

I(nA

)

T~4K

12

43

591317 121

Figure 3.12: Low temperature gate traces of three typical CNT devices. Bymeasuring a gate trace such as presented in this figure single CNTs can be identified byfour-fold shell filling. In some devices the tunnelcoupling is different for electrons andholes. In such case we observe Fabry-Perot type conductance oscillations for holes andCoulomb peaks for electrons. Numbers indicate the number of electrons/holes on thenanotube just after/before the labeled conductance peaks. Temperature was ∼ 4K.

Figure 3.13: SEM image of two CNTs across one trench.

76

3.4. Selecting good nanotube devices

-30

0

30

030

60dI/dV( S)

-30

0

30

-505

dI/dVg (a.u.)

02040

-30

0

30

-505

-30

0

30

dI/dV( S)

dI/dVg (a.u.)

V(m

V)

V(m

V)

V(m

V)

V(m

V)

-8 -4 0 4 8Vg(V)

-6 -4 -2 0 2 4 6Vg(V)

Device A: at least one CNT

Device B: at least two CNTs

Figure 3.14: Low temperature bias spectroscopy of CNT devices. (a,b)In this device the typical checkerboard pattern for a QD in the Fabry-Perot regimeis observed. The pattern is regular and there are no clear features superimposed onit. This is typical for a single small bandgap CNT. (c,d) In this device there are atleast two CNTs across the trench, which can be determined from the extra lines thatform Coulomb diamonds that appear superimposed on the checkerboard pattern. Thederivative of dI/dVg makes this apparent. In both devices the bandgap is on the orderof 30meV, as can be estimated from bias spectroscopy.

77

Bibliography

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[1] W. Kern, The evolution of silicon wafer cleaning technology, J. Electrochem.Soc 137(6), 1887–1892 (1990).

[2] M. Lundeberg and J. Folk, Spin-resolved quantum interference in graphene,Nature Physics 5(12), 894–897 (2009).

[3] X. Liu, Quantum dots and Andreev reflections in graphene, PhD. thesis,Delft University of Technology, 2010.

[4] J. Oostinga, H. Heersche, X. Liu, A. Morpurgo, and L. Vandersypen, Gate-induced insulating state in bilayer graphene devices, Nature materials 7(2),151–157 (2007).

[5] D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, andL. Wirtz, Spatially resolved Raman spectroscopy of single-and few-layergraphene, Nano letters 7(2), 238–242 (2007).

[6] P. Tedrow and R. Meservey, Spin-dependent electron tunneling in super-conducting vanadium and vanadium-titanium thin films, Physics Letters A69(4), 285–286 (1978).


[8] B. Smith and D. Luzzi, Electron irradiation effects in single wall carbonnanotubes, Journal of Applied Physics 90, 3509 (2001).

[9] F. Banhart, The formation of a connection between carbon nanotubes in anelectron beam, Nano Letters 1(6), 329–332 (2001).

[10] A. Haq and O. Meyer, Electrical and superconducting properties of rheniumthin films, Thin Solid Films 94(2), 119–132 (1982).

[11] J. Svensson and E. Campbell, Schottky barriers in carbon nanotube-metalcontacts, Journal of Applied Physics 110(11) (2011).

[12] A. Woolf, An outline of rhenium chemistry, Q. Rev. Chem. Soc. 15(3),372–391 (1961).

[13] CHEMnetBASE, Combined Chemical Dictionary, February 2012,http://www.chemnetbase.com/.

[14] A. Ferretti, D. Rogers, and J. Goodenough, The relation of the electricalconductivity in single crystals of rhenium trioxide to the conductivities ofSr2MgReO6 and NaxWO3, Journal of Physics and Chemistry of Solids26(12), 2007–2011 (1965).

[15] S. Oyama, J. Schlatter, J. Metcalfe III, and J. Lambert Jr, Preparation andcharacterization of early transition metal carbides and nitrides, Industrial& engineering chemistry research 27(9), 1639–1648 (1988).

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79

Chapter 4

Graphene Josephsonjunctions

Organization of this chapterIn this chapter we present results of our experiments on graphene Josephson junc-tions. The goal of our work was to realize a Zeeman π-junction in graphene. Firstwe motivate the use of graphene, and then we propose experiments on grapheneπ-junctions. We present measurement results on three devices, and provide rec-ommendations for future work. We conclude the chapter with a summary of ourresults. The methods of device fabrication are discussed in Sec. 3.2.

Main resultWe find that our graphene Josephson junctions carry supercurrents in in-planemagnetic fields as high as B‖ = 1.5T. In one device we observe a switchingcurrent dependence on B‖ that is typical for a π-junction, but is disputablebecause of obscured Fraunhofer patterns.

4.1 Introduction

Why graphene?Zeeman π-junctions can be characterized by their dependence of switching currenton magnetic field. This requires robustness of switching current to magneticfields. Two mechanisms can cause Cooper pair-breaking by magnetic field, orbitaldepairing and Zeeman depairing (see Sec. 2.6.1). Usually orbital depairing isthe dominant mechanism, and should be suppressed to enhance robustness ofswitching current. This can be done by preventing cyclotron motion of electronsaround the applied field, for example by confining the electron motion in atleast one direction perpendicular to the magnetic field. A consequence of thisis that the in-plane critical field of a film with thickness d < ξ, with ξ thesuperconducting coherence length, can exceed the perpendicular critical field byone or two orders of magnitude [1]. These arguments also hold for Josephsonjunctions, “flat” junctions should be able to carry supercurrents in large in-planefields.

81

4. Graphene Josephson junctions

The reason to use graphene as weak link to make a Zeeman π-junction is itsflatness and its proven ability to carry supercurrents [2]. Experiments on UCFin graphene show that orbital effects are suppressed up to fields B ∼ 10T [3].One other advantage is that graphene is so flat that it is easy to contact withvery thin film superconductors. This is harder to do with for example InAsnanowires, because there the contact has to be at least similar thickness as thenanowire diameter (∼ 50 nm) to avoid contacting problems.

For our purpose graphene is an exceptionally suitable material, because it is avery thin conductor. The thickness of typical sputter- or e-beam-deposited metalfilms is limited to ∼ 6 nm. At lower thicknesses typical films are not electricallycontinuous anymore, and suffer from oxidation. To give one example, Crosser etal. have used thin Ag films as N-part and the minimum thickness they achievedwas ∼ 33 nm. Thinner films were not electrically continuous [4]. With such filmsas weak link between Al contacts, they were able to observe switching current upto B‖ = 0.1T. Unfortunately this field was not large enough to observe a 0− πtransition which was expected to occur in these devices at B‖ = 0.35T. Withour devices we aim to achieve a larger in-plane critical field.

4.2 Proposed experiments on Zeeman π-junctionsin graphene

We propose two experiments for π-junctions on graphene. The purpose of theexperiments is to observe typical signatures of π-junctions. The experiments onlydiffer in the way the contacts are designed, and (by chance) they can be combinedin a single experiment. We have pursued experiment one.

Experiment oneIn this experiment we measure Fraunhofer diffraction patterns as a function ofan in-plane magnetic field. We track the switching current dependence on B‖ atB⊥ = 0. In this way we can find the 0 − π transition field B0−π, and bring thejunction in the π-state. In a diffusive Josephson junction B0−π is determined bythe Thouless energy, which depends on the length of the junction (see Eqs. (2.17)and (2.73)). In the ideal case this occurs across the whole area of the junction ata single B‖ field. At this field sin(2ϕ) components in the current phase relationare dominant, which could be manifested as a period-doubling of the Fraunhoferdiffraction pattern. This is a smoking-gun signature for the 0− π transition. InFig. 4.1a we show the device schematic for this experiment. In panel (b) we showthe Fraunhofer patterns, and in panel (c) a sketch of the field dependence of thefirst and second order components critical current.

Experiment twoIn experiment two the width of the junction is not homogeneous, so that themagnetic field can be tuned such that in one half of the junction the groundstate is at π, while in the other half the ground state is 0. This can be achievedexperimentally by designing contacts as in Fig. 4.1d. In this way a 0−π-junctionis created, in which spontaneously a supercurrent runs around the 0 − π phaseboundary. This vortex generates a fractional flux quantum Φ0/2 [5]. In the

82

4.3. Experimental setup

-6 -4 -2 0 2 4 60

0.5

1

I c

0 JJ0- JJ

B||0

0- JJ

B

-6 -4 -2 0 2 4 60

0.5

1

I c

sinsin 2

B||

0 or JJ

B0 /

0

/2 vortex

0

0

B//,0-

1

I c

-10 JJ JJ

B

(a) (b) (c)

(d) (e) (f)

Figure 4.1: Two configurations for π-junction experiments. (a) Deviceschematic used in our work. (b) Fraunhofer diffraction patterns for sin(ϕ) and sin(2ϕ)current phase relations. Flux is induced by a variation of B⊥, and a change of thediffraction pattern by a variation of B‖. (c) Sketch of dependence of Ic for sin(ϕ) andsin(2ϕ) on B‖. (d) Device schematic for proposed experiment. B‖ can be tuned suchthat the junction is partly in the π-state and partly in the 0-state. (e) Fraunhoferdiffraction pattern in a 0 − π-junction has a dip at B⊥ = 0. (f) In a half 0, half π-junction the ground state holds a Φ0/2 vortex. Diffraction patterns are simulated andnot measured.

presence of such vortex the Fraunhofer pattern changes in a typical way, a diprather than a peak occurs at B⊥ = 0 (see Ref. [6]). This is shown in Fig. 4.1(e,f).In principle a mixed 0 − π-junction can also be created “by accident”, due tomicroscopic variations in the width of the contact that are due to the fabricationprocess [7].

Fraunhofer diffraction patterns have been measured in mixed 0 − π-junctionsbefore [6,8]. The advantage of our approach is that Zeeman π-junctions are fullytunable by magnetic field, and do not rely on a change of temperature or filmthickness to pass from the 0 to the π state.

4.3 Experimental setup

In Fig. 4.2 we give an overview of our measurement setup. We have used a home-made vector magnet that consists out of a superconducting coil that was fixedto the outside of the IVC can, and placed inside the bore of a larger magnetin the helium dewar. This coil is expected to provide a field of ∼ 8mT/A (seeApp. F). We have made special effort on the filtering of our system, by installingcopper-powder filters and RC filters (2nd order, 10kHz cut-off) below the mixing

83


chamber, and providing proper shielding. The resistance of the RC filters is2 kΩ (820Ω/22 nF and 1.2 kΩ/1 nF). Our measurement lines are equipped withπ-filters at room temperature.

B//=1…2T

B =0…20mT

(a) (b)

24xPI

24xCu-P

brass shield

sample

24xRC

300mKpot

(c)

noise at sample

<1 V for each component

trans

fer

f

10kHz 1-10MHz 100MHz-1GHz

>100GHz

RC2 PI Cu-powder

<20pV/ Hz spectral density

1

0 2k /wire

Figure 4.2: Measurement setup and filters. (a) Superconducting coil glued tothe outside of the IVC can to make a vector magnet. (b) The sample is filtered by PIfilters at RT, and Cu-powder and RC filters (sample and RC filters not in picture). Thesample and RC filters are placed in a brass can. (c) Transfer function of used filters.

We have used three different types of superconducting contacts, that will all bepresented. We have made effort, to achieve a fabrication recipe for making closelyspaced sputter-deposited niobium contacts. This is not trivial due to sidewalldeposition during sputtering. In Fig. 4.3 we show typical devices with niobiumcontacts spaced by 280 nm. The contact spacing of our devices was typically200 . . . 300 nm, and the contact width was ∼ 1μm. Contact length varied a bitfrom device to device, depending on the size of the graphene flake, but wastypically 4 . . . 10μm. The area of our junctions is typically A = 8μm · 200 nm =1.6 · 10−12 m2. The required field to put one flux quantum in the junction isΦ0/A ∼ 1mT.

In Sec. 2.6.3 we provided an estimation of B0−π,‖ ∼ 1 . . . 2T for graphene. Nextwe will show how we have optimized our devices to reach a regime where we canobserve supercurrent at such high fields.

84

4.4. Experimental results

410 m

(a)

(b) (c)

Figure 4.3: Graphene Josephson junctions. (a) Sputtered closely spaced niobiumcontacts. (b) Optical microscope image of a typical graphene flake on SiO2. (c) Falsecolor optical microscope image of niobium contacts (yellow) on graphene (blue). Thisis the same flake as in (b).

4.4 Experimental results

In the course of time we have optimized our devices for Zeeman π-junction ex-periments. The start point was using only sputtered niobium. Then we havetried to improve the IswRn product by packing the niobium in a thin titaniumlayer. When we found out that the in-plane critical field was not large enough, weevaporated Ti/V/Ti trilayer films. The resulting IswRn products indicate thatwe have made some progress in these steps, and are presented in App. A.1.1. Wehave found that our Ti/V/Ti films reproducibly gave better results compared toniobium based films.

In this section we will present the results of our experiments on three selecteddevices with different contacts. We start with an overview of the properties ofour superconducting films, and then discuss the graphene devices.

Superconducting filmsOur films are presented in Fig. 4.4. We have used similar films for our devices.Characterization was done in a similar way as presented in App. D, we havemeasured Tc and Bc,‖. We started off with sputtered niobium with a very lowTc ∼ 5.6K (niobium bulk Tc ∼ 9.2K). The low Tc is probably due to poorsputtering conditions in the system used. In panel (b) we show characterizationresults of Ti/Nb/Ti trilayers. We have added Ti to “clean” the contact surfaceof graphene, and to protect Nb from oxidation. In panel (c) we show the resultsfrom our Ti/V/Ti films, that have exceptionally large critical fields exceeding

85


6T. One possible explanation for the large critical field of the vanadium filmis that this film is more “2D”, and our niobium films are more like bulk. Thiscould be confirmed by measuring the angle dependence of the critical field. In afilm Bc,‖ � Bc,⊥, and in bulk superconductors there is no angle dependence [1].Whether our films are bulk or not depends on the ratio between the coherencelength and the film thickness.

(a)

4.5 5 5.5 60

0.5

1

T(K)

V(m

V)

Ti(3)/Nb(100)/Ti(3)nmTc~5.4K, Bc,//>1T

0 0.4 0.80

0.2

0.4

B//(T)

I sw(m

A)

T~350mK

Nb(80)nmTc~5.6K, Bc,//>4.5T

(b)

2.5 3 3.50

1

2

T(K)

V(m

V)

0 2 4 60

0.5

1

I(m

A)

B//(T)

Ti(3)/V(10)/Ti(3)Tc=3.1K, Bc,//=6T

(c)

0 1 2 3 40

30

60

T (K)

R (

)

at B//=4.5T, Tc=1.5K

0

30

60

10 20 30I (μA)

R (

)

T=300mKB//=4.5T

10μm

1μm

B//=

4.5T

d=80nm

Figure 4.4: Characterization of superconducting films. (a) 80 nm niobium filmswith Tc ∼ 5.6K and Bc,‖ > 4.5T. (b) Ti/Nb/Ti trilayers of d ∼ 100 nm thickness havesimilar Tc as the Nb films, but lower Bc,‖. (c) Our Ti/V/Ti films (d ∼ 16 nm) have thelargest critical field Bc,‖ ∼ 6T.

Results on Nb deviceThe results on the Nb device are shown in Fig. 4.5. In panel (a) we find aFraunhofer-like pattern. This pattern is not exactly Fraunhofer, something wehave observed typically in all our devices. Only very few had a close resem-blance to the Fraunhofer pattern. Deviations from such patterns indicate inho-mogeneous current density across the junction, or the presence of vortices in thejunction [6, 9]. In panel (b) we show a typical IV characteristic. We found thatthe switching current is tunable by gate voltage, as shown in panels (c,d,e). Theswitching current is correlated to the conductance of the device, which is typicalfor graphene Josephson junctions [2]. We identify the dip in conductance with theDirac dispersion of graphene. In this device the dip is broad and shallow. Thisis possibly due to non-uniform doping, i.e. the little dip is the average of many“sharp” Dirac dips that reside in a range of gate voltages around Vg = 0. This in-dicates that our device is disordered, which is consistent with an inhomogeneouscurrent density.

During the process of contact deposition it is not unlikely that graphene is dam-aged. Because it is so thin, graphene is very susceptible to dry-etching. Only asecond or so in an argon plasma is enough to etch it away. Although the argonplasma that is used for sputtering niobium is mainly kept at the sputter target,which is at a distance of ∼ 10 cm below the substrate, we can imagine that italso partly etches graphene during the niobium deposition. Because this processdepends on the distance between the substrate and the sputter target, it can

86


give different results in different sputtering systems. Induced superconductivitycan also be reduced by an increased contact resistance due to the incorporationof resist residues in the contact superconductor. Usually an additional titaniumlayer helps to lower the contact resistance, and we have tried this in the secondfabrication run.

4 6 80

2

4

6

I(nA)

V(

V)

Vg=-52V

Vg=0

Isw

-50 0 505.5

6

6.5

Vg(V)

I sw(n

A)

-50 0 5033

35

37

-50 0 500

20

40

Vg(V)

G(e

2 /)

IswRnI=100nA

B (A)

I(nA

)

-0.5 0 0.5 10

5

10

0

2

V( V)

B//=0Vg=0

-10 -5 0 5 10

-5

0

5

I(nA)

V(

V)

(a)

(c)

(b)

(d) (e)

T=350mKRn=745Isw=6nA

Nb

-1

Figure 4.5: Characterization of Nb device. (a) Fraunhofer-like pattern. Insetshows optical microscope picture of the device. (b) IV curve taken at Vg = 0. (c)Switching current is tunable with Vg. (d) Switching current as a function of Vg. (e)The switching current is correlated to the conductance.

Results on Ti/Nb/Ti deviceWe have found one Ti/Nb/Ti device with an exceptionally high switching cur-rent. This device has an interference pattern that closely resembles a Fraunhoferpattern. This is shown in Fig. 4.6a. We made detailed measurements at Vg = 0.In panel (c) we show the IV characteristic and on the inset an optical micro-scope image of the device. We have measured the dependence of the switchingcurrent on the in-plane critical field. Before we start such measurement we firstcharacterize the field dependence of our device. This is shown in panel (b).

Josephson junctions are very sensitive to magnetic field. Since the alignment ofour junction is not perfect, B‖ also induces a small B⊥. We can correct for thisfield by tuning B⊥. In panel (b) we find that increase of B‖ changes the (real)B⊥ = 0 condition towards more negative B⊥. In extraction of Isw(B‖) (panel(d)) we have corrected for this dependence. We find that the switching currentdisappears on a scale of B‖ ∼ 100mT, which is about one order of magnitudesmaller than our estimate of B0−π (Sec. 2.6.3). For this reason we have tried to

87


improve our devices by developing Ti/V/Ti contacts.

-0.5 0 0.5 10

0.5

1

0

80

160dV/dI( ) B//=0

Vg=0

B (A)

I(A

)

-1

-1 -0.5 0 0.5 1

0

40

-40

I (nA)

Rn=60Isw=915nA

Isw

V(

V)

T=350mKVg=0

0 40 800

0.4

0.8I s

w(

A)

B// (mT)

B//(mT)0 40 80 120

-0.3

-0.2

-0.1

0

B(V

)

0

0.5

1V( V)

B//>100mTIsw<10nAVg=0

(a)

(c)

(b)

(d)

Figure 4.6: Characterization of Ti/Nb/Ti device. (a) Fraunhofer-like pattern.(b) Voltage across the junction at low bias as a function of B⊥ and B‖. The current biasis indicated by the dashed green line in panel (a). (c) IV curve taken at Vg = 0. Insetshows an optical microscope image of the device. (d) Switching current dependence onmagnetic field, measured at four points along the red dashed line in panel (b).

Results on Ti/V/Ti deviceWe have thoroughly characterized one Ti/V/Ti device. In Fig. 4.7 we show IVtraces as a function of B⊥ and B‖. In panel (a) we show the IV curve at B = 0.We find Isw ∼ 160 nA. In panel (b) we show the dependence of Isw on B⊥. Wefind a Fraunhofer-like pattern. In the inset we show an optical microscope imageof the device, W × L ≈ 0.5 × 7.5μm2. From scans as in panel (b) we extractIsw(B⊥). We have taken such scans as a function of the in-plane field B‖, andhave plotted Isw(B⊥) in panel (c), for B‖ = 0 . . . 1.3T. Curves are offset forclarity. It is important to notice that we have to shift the curves to correct forthe non-perfect alignment. We do this by tracing recurring features in Isw(B⊥)and shift the curves such that these features are aligned. The features we usedare indicated by dashed green vertical lines.

We find weak oscillations in Isw(B⊥) up to B‖ = 1.3T, which indicates thatthe Josephson effect persists in our devices up to this field. This is encouraging,because it is in the good range for Zeeman π-junction. For this reason we havetaken high resolution scans in the field range B‖ = 0.49 . . . 1.51T. This is shownin Fig. 4.8a.

88


(a)

(b)

(c)

B (A)

I(nA

)

-1 -0.5 0 0.5 10

100

200

300

-400 -200 0 200 400

-40

0

40

I (nA)

T=350mKRn=150Isw=160nAVg=0

0 10 20 30V( V)

B//=0Vg=0

Isw

100

B (A)

I sw(n

A)

0.65

0

200

1.3B

// (T)

V(

V)

Ti/V/Ti

-1 0 1-3 -2 2 3

B

B//

Figure 4.7: Characterization of Ti/V/Ti device. (a) IV curve taken at Vg = 0.(b) Fraunhofer-like pattern. Inset shows optical microscope image of the device (c)From scans as in (b) we extract Isw(B⊥) as a function of in-plane field B⊥. Curves areoffset on the vertical axis, and labeled by the in-plane field on the right y-axis. Dashedlines indicate features that were used to manually align diffraction patterns.

The procedure to extract Isw(B⊥ = 0) as a function of B‖ was as follows: First weshifted the curves horizontally such that recurring features were aligned, secondlywe made a parabolic fit in a fixed fitting range around B⊥ = 0 (red segmentsin Fig. 4.8a), to extract the maximum switching current in this field range. The(manual) alignment by horizontally shifting the curves is of course not withouterror. The parabolic fit helps us to correct for small misalignment errors.

In Fig. 4.8b we show the extracted fit parameters from data in Fig. 4.7, Fig. 4.8aand one other dataset (not shown). Our main finding is a cusp at B‖ = 0.59T,followed by a bump extending to B‖ = 1.32T.

Such “bump” behavior is typical for π-junctions, the cusp corresponds to thepoint where the ground state of the junction changes from “0” to “π” (seeFig. 2.24). This indicates that our junction is perhaps a π-junction with B0−π =0.59T, but the smoking-gun evidence to support a firm conclusion is lacking. InFig. 4.1 we have shown that at the cusp-point we expect to observe a perioddoubling of the Fraunhofer pattern, or in the absence of a sin(2ϕ) contribution,a complete suppression. At the cusp there are still some oscillations visible, theFraunhofer pattern is not completely suppressed. Our S/N ratio is not largeenough to find a period doubling in this range.

89


-0.5 0 0.5 1

40

80

120

B (A)

I sw(n

A)

1

0.49

1.51

B// (T

)

0

0 0.5 1 1.50

40

80

B// (T)

B0- =0.59T

I sw(n

A)

B0- =0.59T

0.5 1 1.50

5

10

B// (T)

B2=1.32T

In~2nA

I sw(n

A)

dataset 1dataset 2dataset 3

(a) (b)

(c)

Figure 4.8: Isw(B⊥ = 0) as a function of B‖. (a) Fraunhofer-like patterns in therange of 0.49 . . . 1.51T. Red lines indicate parabolic fits around B⊥ ∼ 0. (b) Isw(B‖) asextracted from parabolic fits in three different datasets. Error bars are fit residue. Thedashed line is a typical dependence that was taken from Ref. [10], scaled and plottedhere as a guide to the eye. (c) Zoom-in from bump region in panel (b).

90


The interference patterns Isw(B⊥) are not everywhere symmetric, Isw(B⊥) �=Isw(−B⊥), especially above B‖ = 1T. This is shown in Fig. 4.9 where we showa colorplot from which dataset 3 in Fig. 4.8(b,c) was extracted.

0.5 1 1.5-0.8

0

0.8

B (T)

Isw(nA)

B(A

)

B0- =0.59T

90

60

30

0dataset 3

Figure 4.9: Isw(B⊥) as a function of B‖. Colorplot shows the full data from whichthe line marked “dataset 3” was extracted in Fig. 4.8. The bump at B⊥ ∼ 0 is clearlyvisible. The cusp position is indicated by the vertical arrow.

This is a complication that makes the interpretation of our results harder. If weneglect these complications for a moment, we can use Eq. (2.73) to estimate theThouless energy of our device: ETh ∼ 4.3μeV. The dimensions of our deviceare L × W ≈ 0.5 × 7.5μm2, and that gives a diffusion constant D ∼ 16 cm2/sfollowing Eq. (2.17). The order of magnitude of our device dimensions and Tc

of the leads is comparable to those of other experiments reported on grapheneJosephson junctions [11, 12]. Those junctions were in the diffusive regime, andvalues of resp. ETh ∼ 80μeV and D ∼ 110 cm2/s are found in Ref. [12], andETh ∼ 260μeV and D ∼ 124 cm2/s are found in Ref. [11]. We find valuesthat are at least one order of magnitude smaller, which possibly indicates thatour graphene is more “dirty” compared to those experiments. However morecharacterization and further experiments should be done to confirm this.

Symmetry breaking due to trapped fluxIn the presence of time-reversal symmetry (TRS) Fraunhofer-patterns have to besymmetric under reversal of the direction of magnetic field. In our experimentTRS is broken by B‖, but this does not directly alter the electron trajectories,since these are confined in the direction parallel to the field. The likely candidateto break the symmetry of electron trajectories is trapped flux due to a magneticvortex near the junction. The effect of flux trapping can be calculated [6], andresults in a breaking of the symmetry of the Fraunhofer pattern.

In our experiment we observe asymmetric patterns at B‖ > 0, which indicatesthat flux is trapped during the field sweep. We have observed that interferencepatterns can be reproducible/stable during days, but can also change. This in-dicates that flux trapping is changing in time, and vortices move through thejunction or contacts. This obscures our measurements and was the main moti-vation to not continue the experiment at the time. In Fig. 4.10(a,b) we showtypical interference patterns.

91


B||

B

-0.5 0 0.5 1B (A)

0

B//(T)=490mT, three datasets

10

I sw(n

A)

-0.5 0 0.50

50

B (A)

B//(T)=80mT, two datasets

I sw(n

A)

(a) (b)

(c) (d)

Figure 4.10: Effect of trapped flux. (a) Three datasets show reproducible asym-metric interference patterns, indicative for trapped flux. (b) Two datasets taken witha couple of days in between show a changing interference pattern, indicative for movingflux. (c) Sketch of moving vortices in left contact, and pinned vortices in right contact.(d) Figure adapted from Ref. [13] shows ∼ 40× 40μm2 holes in SC film to trap flux.

Alternative explanation for cuspAn alternative explanation for the cusp is that it is due to similar Fraunhoferdiffraction as observed with B⊥, but now with B‖. This is in principle possible ifthe alignment is not perfect, as is shown in Fig. 4.11. In that case the first node ofinterference in the plane L× dZ is at B‖ = 0.59T, which corresponds to an areaL×dZ = φ0/0.59T ≈ 7.5μm×0.5 nm. This corresponds to α = arctan 0.5/500 <0.1�. It is unlikely that our alignment is that good, a misalignment of ∼ 5� orso is more likely. In that case dZ ∼ 40 nm and the first node is expected at7.5×0.04μm2/φ0 ≈ 150mT. This implies that we should have observed a coupleof Fraunhofer diffraction nodes in the range B‖ = 100 . . . 500mT. We find thatwe cannot simply explain the cusp with Fraunhofer diffraction due to B‖, becauseit would imply a misalignment error that is much smaller than we expect.

There is a straightforward way in which Fraunhofer interference due to B‖ canbe distinguished from Zeeman-π interference, simply by mounting the junctionsuch that B‖ is pointing along the direction of the current. In Fig. 4.11a thiscorresponds to rotating the device by 90� in the xy-plane. In panel (b) and (c) weshow the orientation of our device, and the proposed orientation. If the cusp isdue to Zeeman-π interference it should not depend on a rotation of the in-planefield, but the Fraunhofer diffraction does change with rotation of field.

92

4.5. Conclusion

B B

LW

x yz

dZ B

B//

B

B//

(a) (b) (c)

Figure 4.11: Junction alignment. (a) A sketch of the alignment of our device. Wehave tried to minimize the misalignment angle α, to keep the area L × dZ minimum.Dashed arrows indicate supercurrent moving in the junction plane (grey plane). (b)Orientation of Ti/V/Ti device. (c) Proposed orientation to reduce potential Fraunhoferdiffraction by B‖.

4.5 Conclusion

In our effort towards Zeeman π-junctions in graphene:

� We characterized different SC contacts and found that Ti/V/Ti repro-ducibly gave the best results.

� We realized a Josephson junction with an in-plane critical field as high asB‖ = 1.5T.

� We observed a switching current dependence on B‖ that is expected forπ-junctions.

� The observed B0−π = 0.59T, which has the right order of magnitude as isexpected based on Eq. (2.73), and assuming our graphene is “dirty”.

� We observed asymmetric and slowly drifting diffraction patterns indicativeof the presence of trapped flux.

RecommendationIf this experiment is to be continued, effort should be made to minimize fluxtrapping. This can be done by reducing the width of the leads and contacts tographene to ∼ 100 nm, this shifts the field at which a single flux quantum can beaccommodated to B⊥ > 50mT, which is about ten times above the fields neededto observe Fraunhofer patterns. Large areas of SC film should be penetrated withholes to trap vortices. This is a standard procedure in SC (quantum) electronics.We have adapted a figure from Ref. [13] in Fig. 4.10d to illustrate this.

The junction should be mounted such that the in-plane field is pointing alongthe direction of the supercurrent.

93

Bibliography

Bibliography



[3] M. Lundeberg and J. Folk, Spin-resolved quantum interference in graphene,Nature Physics 5(12), 894–897 (2009).

[4] M. Crosser, J. Huang, F. Pierre, P. Virtanen, T. Heikkila, F. Wilhelm,and N. Birge, Nonequilibrium transport in mesoscopic multi-terminal SNSJosephson junctions, Physical Review B 77(1), 014528 (2008).

[5] L. Bulaevskii, V. Kuzii, A. Sobyanin, and P. Lebedev, On possibility ofthe spontaneous magnetic flux in a Josephson junction containing magneticimpurities, Solid State Communications 25(12), 1053–1057 (1978).

[6] D. Van Harlingen, Phase-sensitive tests of the symmetry of the pairing statein the high-temperature superconductors, Reviews of Modern Physics 67(2),515 (1995).

[7] S. Frolov, Current-Phase Relations of Josephson Junctions With Ferromag-netic Barriers, PhD. thesis, University of Illinois at Urbana-Champaign,2005.

[8] M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D. Koelle, R. Kleiner,and E. Goldobin, 0-π Josephson tunnel junctions with ferromagnetic barrier,Physical review letters 97(24), 247001 (2006).

[9] A. Barone and G. Paterno, The Josephson Effect, Physics and Applications(1984).


[11] C. Chialvo, I. Moraru, D. Van Harlingen, and N. Mason, Current-phaserelation of graphene Josephson junctions, Arxiv preprint arXiv:1005.2630(2010).

[12] G. Lee, D. Jeong, J. Choi, Y. Doh, and H. Lee, Electrically Tunable Macro-scopic Quantum Tunneling in a Graphene-Based Josephson Junction, Phys-ical Review Letters 107(14), 146605 (2011).

[13] H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. Bialczak, M. Lenander,E. Lucero, M. Neeley, A. O’Connell, D. Sank, et al., Improving the coher-ence time of superconducting coplanar resonators, Applied Physics Letters95(23), 233508–233508 (2009).

94

Chapter 5

Vibrating suspended cleancarbon nanotube Josephsonjunctions

In this chapter we will discuss the results from the experiments on two suspendedCNT Josephson junctions. We have investigated the effect of mechanical reso-nance on the Josephson junction and compared signals when the contacts werein the normal or superconducting state.

5.1 Introduction

Focus of experimentsWe study mixing signals that are enhanced by oscillating Josephson currents.In our devices (see Fig. 5.1) oscillating currents can be induced with the ACJosephson effect. These can be in the same frequency regime as mechanicalmodes of the CNT. Due to capacitive coupling between CNT and gate, CNTdisplacement will change the induced charge and in this way also cause oscillatingcurrents.

21

Vg

Ze VDC

y

VRF

insulator

gate

Figure 5.1: Schematic picture of our device. The mechanical resonator is asuspended CNT on top of superconducting contacts. The device is biased through anunknown impedance Ze.

95

5. Vibrating suspended clean carbon nanotube Josephson junctions

We will investigate if such mechanical currents can be detected by mixing withoscillating Josephson currents. In this introduction we will first present our mainresults, with the aim to prepare the reader for this chapter.

Main resultsWe do not observe any signal from the mechanical resonator without driving itby RF signals. When we drive it, we find a mechanical signal that appears inthe same power and frequency regime as where Shapiro steps are observed. Wehave studied this signal with the aim of gaining understanding of its origin, andmore specifically if it is related to: 1. The presence of superconductivity in theleads and 2. Josephson dynamics. We will do this by distinction between mixingand rectification, and comparison of signals in the superconducting and normalstate. We have found the following results in two devices.

We find that the mechanical signal is enhanced when the leads are superconduct-ing. This is most clearly indicated by the dependence on field and temperature.The mechanical signal vanishes above the critical field (Bc,‖ = 2.1T), and alsoabove the critical temperature (Tc ∼ 2.4K). To attribute the enhanced mechan-ical signal to mixing with Josephson currents (Josephson mixing) is not trivial.We find that the signal is due to mixing by studying the gate and bias depen-dence. We find that the mixing signal is enhanced at low field (in comparison tohigh field) and at I ∼ 0 (in comparison to I � Isw). This enhancement allows usin one device to measure the amplitude of the CNT resonator down to ∼ 3 pm,which is on the order of twice the zero point motion, and on the order of thethermally driven motion of the resonator.

Our strongest indication for Josephson mixing is a change of the signal sign (aresonance lineshape is flipped upside down), that occurs as a function of magneticfield.

We interpret the sign change as a consequence of Zeeman splitting of Andreevbound states, that results in a 0−π transition above a certain magnetic field. Inthe π-state the current through the junction is reversed, which results in a signchange of the mixing signal.

We find that mixing with Josephson current is the most likely cause of the en-hanced mechanical signal in our devices. The mixing signal allows us to retrievethe sign of a part of the supercurrent spectrum that is resonant with mechani-cal vibrations. If our interpretation is correct, the experiment makes it possibleto perform a phase-sensitive measurement of supercurrent, using only a singleJosephson junction.

We find a “foot-like” structure in the field dependence of the switching current,which is possibly an indication of a 0− π transition as well.

In one device we find at low current bias and low power (on Shapiro plateaus), anon-trivial power dependence that we cannot explain by gate force driving andmixing with normal currents. Now that we have presented the main results ofour experiment it is possibly easier to follow the structure of this chapter.

96


Organization of this chapterIn Sec. 5.2 we present our experimental setup. In Sec. 5.3 we present the basicelectrical and mechanical characterization of our devices. In Sec. 5.4 we presentour main finding: A signal that appears when the AC Josephson effect is onresonance with a mechanical mode. In the remaining part of this chapter westudy the dependence of this signal on: Gate voltage (Sec. 5.5), driving power(Sec. 5.6), magnetic field (Sec. 5.7) and temperature (Sec. 5.8). Throughout wewill compare measurements taken at positive and negative current bias, and inthe superconducting and normal regime.

In Sec. 5.9 we will study the mechanical resonance at low power and low currentbias, in the regime where there are Shapiro plateaus. In Sec. 5.10 we present anoverview of the observed effects, and present our conclusions.

5.2 Experimental setup

Our experiments are performed in a Leiden Cryogenics dilution fridge with abase temperature of ∼ 80mK. The large base temperature was probably dueto a broken valve in the mixture circuit that limited the flow. Our fridge isequipped with similar filtering as presented in Sec. 4.3. We have used home-built electronics for low-noise measurements. RF signals were applied with aRohde Schwarz SMR40 source. For measuring mechanical resonance we haveused a computer controlled ADwin Gold system. Frequency sweeps and dataacquisition were synchronized. The fridge is equipped with a superconductingmagnet. We have applied fields up to B ∼ 4.5T. The direction of the magneticfield is in-plane with the contacts and parallel to the direction of the trench. Inthis case the magnetic field is mostly perpendicular to the suspended CNT.

In Fig. 5.2 we present our experimental setup. In panel (a) we show the cold fin-ger of the dilution fridge, that is equipped with 36 filtered lines. Before cooldownwe mount a copper can (not shown) on the threading beneath the Cu-powderfilters. The RF cable is mounted on a 10 dB attenuator at the cold plate. An-other 20 dB attenuator is present at the 1K plate (not shown). The RF cableis wound a couple of times around the copper powder filters (to benefit fromlossy transmission line attenuation at high frequencies) and fed through a tinyhole to the device. In panel (b) we show how our chip carries are mounted. Theantenna is freely suspended above the chip carrier. In panel (c) we show a SEMpicture of the chip carrier, taken after measurements were finished. We have usedaluminum bonding wires between the gold plates of the carrier and the rheniumcontacts. The bonding wires become superconducting below 1.2K. Each contactis split at the gold plates, to allow for four-point measurements.

97


(b)

(c)

cells

RFantenna

Vsd

Isd Vg

(a)

Cu powderfilters

RC filters

coax line

chipcarrier

20mKcold plate

10dB

Vrf

LPF LPF LPF LPF LPF

Vrf

Figure 5.2: Experimental setup. (a) Cold finger with filtering and chip carriers.(b) Chip carriers and RF antenna. (c) SEM image of chip carrier (taken after measure-ments were finished) and schematic representation of the measurement configuration.

98


Technical detailsData acquisition for low-noise resonant lineshape measurements was done witha single-datapoint integration time of 1.5ms. The bandwidth of the filters andvoltage amplifier is 10 kHz at a gain of 103, which we have used for almost allmeasurements. We have taken 150 datapoints per trace, which sets the data-acquisition rate for a single linetrace at 4.4Hz. This is above the 1/f cornerfrequency of our amplifier, which is at 2Hz. The 1/f noise is not present in asingle trace, but appears in the offset between traces. This implies that we canintegrate many lineshapes without suffering from 1/f noise in the lineshape. Wefound that 50Hz interference was the main source of noise in our setup, and haveoptimized the timing between different traces to average out 50Hz components.The noise level of the amplifier used is 2 nV/

√Hz at 1 kHz. The noise current is

5 fA.

The RF line was equipped with DC blockers to reduce 50Hz interference.

SEM images deviceIn Fig. 5.3 we show SEM images (taken after measurements were finished) atseveral different magnifications, zooming in on a device with one suspended CNT.

(a) (c)(b)

(d) (f)(e)

Figure 5.3: SEM images of device structure. (a-f) Zooming in on the suspendedCNT (device one). Dashed boxes indicate the zoom region. All SEM images were takenafter the measurements were finished.

Disclaimer magnetic fieldUnfortunately we had to recalibrate the magnetic field scale of the datasets pre-sented in this chapter and Sec. A.2, after the measurements were finished. Webelieve that the magnetic field that was logged during the measurements, andused as setpoint for the magnet power supply, was not the correct value, butwas overestimated by a factor 1/0.765. We were unable to trace this factor to a

99


bug in the programming code, and suspect that an improper setting of the T/Afactor in the magnet power supply caused the initial miscalibration.

Unfortunately we have not logged the magnet current, so we were unable toconfirm this hypothesis. We have recalibrated the magnetic field scale, using thecritical field of the rhenium contacts. This was done as follows: In App. D wereport on the in-plane critical field of a 20 nm Re film, which was measured ina 300mK setup using the “old” Oxford power supply. We found Bc,‖ = 2.1T.The datasets presented in Ch. 6 were acquired prior to those in this chapter,using the “old” Oxford power supply in a dilution fridge. We found Bc,‖ = 2.1T(see Fig. 6.7) of the contacts in device one, which is in good agreement with theexpected critical field. In between the experiments presented in Ch. 6 and thischapter, there occurred a technical problem with the “old” Oxford power supply,and it was replaced with a “new” power supply, with which the datasets in thischapter were taken. We expect that the T/A factor in this “new” power supplywas not properly set. We have corrected for this afterwards by multiplying themagnetic field values by 0.765, such that the leads critical field of device one wasscaled to Bc,‖ = 2.1T.

5.3 Characterization

In this section we present SEM images of our devices and the results of their elec-trical characterization. We extract physical parameters like resonance frequency,level spacing and coherence length, and use these parameters to determine thetransport regime. Here we also present the temperature of the switching currentIsw. Our purpose is to find a good regime to study mechanics.

Main resultIn two similar devices we find gate-tunable supercurrent of ∼ 1 nA. The conduc-tance (of ∼ 4e2/h), and bias spectroscopy indicate our devices are in the ballisticregime. We find gate-tunable mechanical modes at ∼ 1GHz. Switching currentsare observed below ∼ 700mK. A proximity-induced gap of 75μeV indicates ourjunctions are in the short-junction limit.

5.3.1 SEM images

In Fig. 5.4 we show SEM images of our devices taken after measurement. We havefound only one suspended CNT across trenches in device one and device two. InFig. 5.4c we can see that there is an additional CNT present in device one, thatis suspended between a contact and the oxide surface. The SEM images showthe necessity of preselection as described in Sec. 3.4, because we can clearly seethat the CNT is contaminated by the SEM (Fig. 5.4f). After being in the SEMbeam for ∼ 2h the suspended part has increased in diameter by a factor ∼ 10,probably due to contamination that is deposited by the electron beam [1]. Duringthis time we were tuning the SEM to get good images. Device two snapped aftermeasurement, possibly by electric discharge prior to mounting in the SEM.

100

5.3. Characterization

(a) (b) (c) (d)

Lcnt 210nm

device 1

(e) (g)

Lcnt 375nm

device 2(f)

after SEM exposure

Figure 5.4: SEM images of our devices. (a-f) Images taken from device one. (a)Side view. (b) Top view and CNT length. (c) Zoom-in side view. (d) Top view ofcatalyst and trenches. (e) Side view of catalyst, CNT and trench. Position of CNT isindicated by the arrow. (f) After SEM exposure. (g) SEM image of device two. AllSEM images were taken after measurement.

Table 5.1: Device properties.

Eg α ΔE Γ L Tl Tr 4Δp Tp

(meV) (10−3) (meV) (GHz) (nm) (μeV) (mK)1 54 6 7.5 37 220 0.86 0.93 320 5302 25 6 4.6 20 360 0.87 0.94 280 460

5.3.2 Electrical and mechanical characterization

We have found mechanical modes on the order of 1GHz in both devices. Bothdevices can carry a supercurrent on the order of 1 nA, on the hole side as wellas on the electron side (see Sec. A.2.1 for a wider range characterization). InFig. 5.5 we present the gate dependence of the conductance, bias spectroscopy,mechanical resonance frequency and switching current. We find that the observedresonance signal is tunable by gate, which indicates that it is a mechanical modeof the nanotube. The tension in the CNT is increased by a Coulomb force that isdue to the DC component of Vg. This results in resonance frequencies increasingwith the gate voltage.

The regularity of the checkerboard patterns in Figs. 5.5(b,c, top), and the con-ductance being very close to 4e2/h indicate ballistic transport through four chan-nels. From these patterns we have extracted the alpha factor α, tunnel rate Γ,level spacing ΔE, CNT length L and barrier transparency Tl,r, all in the openregime where the conductance is dominated by Fabry-Perot interference. Fromtwo datasets (Figs. A.2b and A.3b) we have extracted the value of the CNTbandgap, Eg. Data is presented in Tab. 5.1.

We find that the CNT length extracted from ΔE is of the same order as thelength found in SEM inspection. We find that the resonance frequencies havethe right order of magnitude if compared to the length and predicted frequency

101


Device 1

Vg(V)

-12 -6 0 6 12024

G(e

2 /h)

1332.5

1342.5

f r(M

Hz)

1322.5

2

3

4

-30

-15

0

15

30

V(m

V)

G(e2/h)

Vg(V)

0.6

1.1

1.6

I sw(n

A)

-6 -5.25 -4.5 -3.75

Device 2

916

918

920

922

f r(M

Hz)

-30

-15

0

15

30

V(m

V)

G(e2/h)

3

4

2

Vg(V)

0.85

1.05

1.25

I sw(n

A)

-9.75 -9 -8.25 -7.5

Device 1

Device 2024

G(e

2 /h)

-6 -4 -2 0 2 4 6

(a)

(b) (c)

I=7nAI=-7nA

Q~30k

I=7nAI=-7nA

Q~50k

Figure 5.5: Characterization. (a) Gate traces measured in the superconductingstate with a voltage bias of Vsd ∼ 50μV (two point measurement). (b,c, top) Biasspectroscopy measured in the superconducting state, using current bias (four point mea-surement). Conductance vs. voltage bias data has been extracted using a histogramtechnique. Both datasets show the checkerboard pattern that is characteristic for quan-tum dots in the Fabry-Perot regime. (b,c, mid) We have measured the resonancefrequency of one mechanical mode, using a method that will be described in the follow-ing section. (b,c, bottom) Tunable switching currents on the order of Isw = 1nA areobserved in both devices.

in Tab. 2.1. In device one we find a lower resonance frequency than predicted,920MHz/1.7GHz and device two has a higher resonance frequency than pre-dicted, 1.3GHz/540MHz. Here we have assumed that we are detecting the fun-damental bending mode of the CNT. It might as well be that for example in devicetwo we detect the first harmonic. Our focus is not on understanding the modesof the resonator, but rather on the mechanism that generates the measurementsignal due to a particular mode.

From a fit to Eq. (2.13) of a single conductance period in Fig. 5.5a, we canestimate the tunnel barrier transmission coefficients for each device. We findtransmission coefficients of ∼ 0.9. In the regime of T ∼ 1 the tunnel rate Γ is

102


determined by the time τ = h/ΔE it takes an electron to travel through bothtunnel barriers.

By the method discussed in Sec. 2.1.3 we find in both devices a tunnelrate on theorder of 30GHz, implying that electrons hop on and off the nanotube roughly 102

times during a single period of a mechanical oscillation. We also notice that dipsin the resonance frequency that usually occur when tuning through a conductanceoscillation, are absent in the gate range of our choice. The reason for this is thatin the Fabry-Perot regime the charge changes continuously as a function of gate,while the dips observed in resonance frequency in other experiments [2–4] arecaused by spring constant softening that is due to steps in charge that occur inthe Coulomb blockade regime [5, 6].

At low gate voltage, Coulomb blockade is present in our devices. But we operatein a gate range where no features of Coulomb blockade are present (Fig. 5.5b-c),and where the charging energy U is very small compared to the other energyscales. We conclude that both devices are in the strong coupling regime whereΓ � Δpr, U . As discussed in Secs. 2.1.4 and 2.2, in this regime Cooper pairscan tunnel from one superconductor to the other, through ABS in the CNT.Cooper pair transport is measured as a supercurrent, by applying a current biasand sweeping the current source to increasing values. The value of the currentat which the junction switches to a voltage-carrying state is called the switchingcurrent. For short, ballistic junctions, theory predicts the maximum value ofsupercurrent (or critical current) in a system with four conductance channels:Ic = (πΔ/e)4e2/h. If this theory holds for CNTs, we would expect Ic,1 = 70nAand Ic,2 = 80nA in our devices.

In Fig. 5.5(b,c, bottom) we present the gate dependence of the switching current.The switching current is correlated to the conductance. We find that for deviceone the switching current is 1.05nA, and can be tuned by gate with ±0.2 nA.In device two the switching current is 1.1 ± 0.35 nA. For device one we findIswRn = 8±0.5μV and for device two we find IswRn = 7.8±1μV. The maximumproximity induced gap can be calculated with the expression from BCS theory [7]:eΔ = 1.76kBTc = 360μeV. In Sec. 2.2.4 we found the theoretical prediction forthe IcRn product: IcRn = πΔ/2e = 570μeV. The measured IswRn product isroughly two orders of magnitude smaller than this upper bound. Usually this isattributed to the influence of the electromagnetic environment [8].

5.3.3 Temperature dependence of switching current

In Fig. 5.6 we show the temperature dependence of the switching current Isw andthe retrapping current Ir in device one. A supercurrent branch is observed atT < 770mK, and hysteresis is observed at T < 450mK. In the regime whereT < 400mK, the Isw(T ) has an exponential decay. This has been predicted forballistic systems [9]. Similar data (measured on diffusive graphene junctions)can also be fit with a model for diffusive junctions [10], and hence we cannotconclude that device one is ballistic based on this data only. At T ≈ 80mKwe find (at this gate voltage) Isw = 1.1 nA and Ir = 0.52 nA. We can usethese values to estimate the (Josephson) quality factor of device one through

103


-2 0 2-20

0

20

40

I(nA)

VV

)

0 0.5 10

0.5

1

T(K)

I sw(n

A)

450mK770mK

Ir

Isw

IswIr

Figure 5.6: Temperature dependence of switching current (device one). (a)Switching current Isw and retrapping current Ir as a function of temperature. (b)Three IV ’s taken at 80mK, 450mK and 1K. All data was taken at Vg = −8.715V.The normal state resistance at T = 1K: Rn = 7.9 kΩ.

Eq. (2.35): Q ≈ 4Ic/(πIr) ≈ 2.7 (we have taken Ic = Isw). Now we can useEq. (2.32) to extract the capacitance of the Josephson junction and the plasmafrequency: CJJ ≈ 35 fF and ωp/(2π) ≈ 1.6GHz. We note that this is not the“direct-local” capacitance between the metal islands on each side of the trench,but the total relevant “participating” capacitance. This is determined by thecapacitance of the part of the circuit that is relevant until a distance wherethe wiring inductance cuts of the signal at the characteristic frequency. UsingEq. (2.33) we find ωc/(2π) = 4.2GHz.

5.3.4 Proximity induced gap

In Fig. 5.7 we extract the proximity induced gap. We have plotted the voltagehistogram of three IV curves that were extracted during a measurement thatwill be presented later, in Sec. 5.7.7. These IV curves have very low noise andallow us to see weak changes of the slope. The histogram is a way to enhancethe visibility of these features. It is similar to voltage bias-spectroscopy, but ina current biased configuration. An advantage here is that the voltage across thejunction is directly measured. When there are many counts at a specific voltage,dV/dI is flat, which corresponds to a high conductance.

At B = 0 we observe two conductance peaks at opposing current bias. In SIStunnel junctions these kind of peaks are due to the quasi-particle DOS in the SCleads. This has also been depicted in Fig. 2.7c. In a voltage bias-spectroscopyexperiment the distance between the conductance peaks corresponds to 4Δpr. Inour experiment the device is current biased. The voltage bias-spectroscopy isdone numerically by making a histogram of the IV traces. In this way we extracta gap of 4Δpr = 320μV for device two, and in a similar dataset (not shown) fordevice one we find 4Δpr = 280μV. Using the BCS expression Δ = 1.76kBTc [7],we extract the proximity critical temperature Tpr ∼ 0.5K. Experimentally wefind supercurrent at 750mK, which is of the same order of magnitude as Tpr.

The hypothesis that we measure the proximity gap is supported by its suppressionat high field. However, it is a bit surprising that the gap is still visible at B =

104

5.4. Shapiro steps at the mechanical resonance frequency

1.8T. The Pauli-limit gives an expression for the field at which Cooper pairsare broken by Zeeman splitting [7]. Using Eq. (2.68) we find Bc,Pauli ∼ 0.9Tusing Tpr ∼ 0.5K. Breaking of Cooper pairs due to Zeeman splitting can besuppressed by spin-orbit interaction [11], which could play a role here as well. InSec. 5.7 we will make a case for the presence of AC supercurrent all the way upto B‖ = 2.1T, where the SC in the leads disappears. If this model is applicablehere, it implies that our estimation of Δ is at least a factor of two off and/or thatspin-orbit interaction in the CNT prevents breaking of Cooper pairs.

-0.5 -0.25 0 0.25 0.51

1.2

1.4

V(mV)

coun

ts/c

ount

s V=

0

320 V 0T

1.8T

2.4T

V

I

n=3n=1n=1

(a) (b)

Figure 5.7: Voltage histogram device two. (a) Illustration of a histogram extrac-tion from an IV curve. In flat parts (high conductance) there are more measurementpoints in a single bin than in steep parts. (b) Histogram of an off-resonant IV curvefrom the dataset in Fig. 5.30, that will be discussed in Sec. 5.7.7. At B = 0T there isa gap of 320μeV, which is suppressed at B = 2.4T.

With Eq. (2.18) we estimate the coherence length of Cooper pairs: ξ = 3.6μm.Since our devices are roughly one order of magnitude shorter, they are in theshort-junction regime. Additionally we observe Fabry-Perot oscillations and G ∼4e2/h, therefore our devices are ballistic Josephson junctions in the short-junctionlimit. In this regime there can exist two ABS per conductance channel [12]. Sincethere are four channels in our device there can be maximum eight ABS present.

5.4 Shapiro steps at the mechanical resonancefrequency

In this section we investigate Shapiro steps induced by microwave signals thatare resonant with mechanical modes in the CNT. We will show how the resonatorsignal appears in our measurements and use the signal to extract the linewidthof our resonators. The linewidth sets a lower limit to the quality factor of theCNTs, which we will also present. We will first present data of a first-generationdevice that has not been discussed in this thesis before, but in which the effectwas first (and most prominently) measured. Secondly we will present data fromdevice one and two. These devices are from the second generation and we haveinvestigated those in detail.

105


Main resultIn three devices we observe a signal from the mechanical resonator as a currentshift. The signal is present in the power regime where Shapiro steps are. We findFano-lineshapes and bias dependence that are typical for mixing signals.

5.4.1 Current shift on resonance in first generation device

The first generation of devices was produced on different wafers and with differentsuperconducting contacts than used for the two devices that will be discussed indetail. We used layered wafers with gate-layers (similar to those reported inRef. [13]), and rhenium that was grown in a MBE system at UCSB (similar tofilms reported in Ref. [14]). The batch of layered wafers was not good, there weremany problems with leakage currents. This prevented good characterization andstudy of these devices and we started fabricating our second generation devices.The quality of the first generation devices was sufficient to show the main featureof this chapter, a current shift on resonance. Here we will just present this signalwithout going into details yet.

The signal is obtained as follows: We drive the system at a frequency fn, andpower P using a microwave antenna and RF source. We take IV curves whilestepping the frequency to fn+1 = fn+Δf , where Δf � Γr and Γr is the linewidthof the resonator at power P . We step the frequency through the mechanicalresonance.

Such a dataset is presented in Fig. 5.8. It shows a shift of the pattern thatappears when the RF frequency is on resonance with a mode in the mechanicalresonator.

In colorscale we have plotted dV/dI. Sharp lines indicate jumps in the IV curvesthat are due to the AC Josephson effect. In between two sharp lines are Shapirosteps. Most Shapiro steps are smoothed out and not clearly visible, somethingwe typically observe at f < 1GHz. The position of Shapiro steps changes as afunction of frequency. This is probably due to the impedance of the RF line, thatis not constant in this frequency range.

The feature of interest is a shift of the pattern at the position indicated by thearrow. This position can be tuned by a gate (data not shown), and is due toa mechanical mode of the CNT. The mechanical nature of the signal motivatedus to reproduce the effect in a new generation of devices that were fabricatedon better wafers. We have also made shorter suspended tubes, such that theresonance frequency was increased to ∼ 1GHz. In this regime Shapiro stepsare more pronounced. In the remainder of this chapter we will only discuss thesecond generation devices, which we have called device one and device two.

5.4.2 Shapiro steps and mechanical resonance

We have measured similar current shifts as presented in Fig. 5.8b in device oneand device two. First we will present in detail in which part of the parameterspace we take our data. Because we are searching for mixing signals, we tune

106


1

6

-1

0

1

I (nA

)

251.5 252.5 253.5f (MHz)

dV/dI(a.u.)

SCCNT

SC

CNT mechanical mode

(a)

(b)

Figure 5.8: Shapiro steps and mechanical resonance in early device. (a) SEMimage of one of the early-stage devices. (b) When the RF source hits the mechanicalresonance all IV curves are shifted. The mechanical resonance is at fr ∼ 252.5MHz.

the gate voltage to a point where the transconductance, dG/dVg is large. Therewe can measure switching current. If we turn on the RF source we find Shapirosteps, as shown in Fig. 5.9.

-1.5 0 1.5-808

1624

I(nA)

V(

V) -53dBm

-68dBm

-39dBm

f=919.14MHz

(a) (b)

-2 -1 0 1 2

-10

0

10

I(nA)

V(

V)

RF off RF on

Figure 5.9: Supercurrent and Shapiro steps in device one. (a) IV curve showssupercurrent. (b) Shapiro steps appear when the RF source is switched on.

We confirm that we are dealing with Shapiro steps by taking IV curves as afunction of power. We pick an RF frequency close to the mechanical resonance.This is presented in Fig. 5.10a, top panel. The frequency of the RF source was

107


set 400 kHz below the mechanical resonance. Shapiro steps are present in ourdata and the characteristic Bessel-function power dependence is also observed.While sweeping the RF power from low to high, we decrease the step size suchthat the sweep follows

√P (dBm).

In the bottom panel of in Fig. 5.10a we have plotted dV/dI in the same greyscaleas the top panel. But now instead of stepping the power for a fixed frequency, westep the frequency for a fixed power. We notice a slight kink in the Shapiro steplines. This kink is due to mechanical resonance, similar (but less pronounced) asin Fig. 5.8.

In Fig. 5.10b we present a zoom in on the boxed area. The inset curve showsthe envelope of the shift, and has been taken from Fig. 5.10d. In Fig. 5.10c weshow the measured voltage (with off-resonant voltage subtracted). The currentshift appears as a peak in voltage across the whole bias range. The height ofthe peak depends on the local slope of the IV curve. A linecut from a similardataset taken at low power (indicated by donut) has been plotted in Fig. 5.10d.A fit to a Fano lineshape is indicated by the grey line. We extract a qualityfactor Q = fr/Γ = 520k. This is not necessarily the intrinsic quality factor ofthe resonator, but rather a measure of the linewidth Γ. Because we do not knowbeforehand if our device is in the linear regime, the measured linewidth might bebroadened by mechanical nonlinearities.

The position of the current shift can be tuned with the gate voltage, whichindicates it is due to a mechanical mode of the CNT. The gate dependence ofthe mechanical resonance frequency presented in Fig. 5.5b was measured in thisway, and will be discussed in detail in Sec. 5.5.

Size of the signalFor technical reasons we have chosen the operation point for further experimentsat I = 7nA, because at this point there are no Shapiro steps, and dV/dI isconstant. We can estimate the magnitude of the mechanically generated DCcurrent as ΔV/Rn = 200 nV/8 kΩ = 25 pA. This corresponds to a ∼ 2.5% shiftof the switching current, which explains why the shift is only weakly visible inFig. 5.10(a,bottom). In Sec. A.2.2 we will show how the signal depends on thelocal slope, and compare its dependence on dV/dI at I ∼ 0 to that at I ∼ 7 nA.

Power dependenceIn Fig. 5.11 we present the measured lineshapes at three values of power. Theworking points at which these curves are taken are indicated by a star, trapezoidand donut in Fig. 5.10(a,top). The lowest power at which it is still convenient tomeasure (limited by integration time) is for this device reached at −63 dBm. Atthis power the lineshape was measured by integration of 6200 frequency sweeps of150 points. Data acquisition was done at 4.4Hz (1.5ms/pt), the total integrationtime was 23 minutes.

Data from device twoIn Fig. 5.12 we present data for device two. In Fig. 5.12a we plot the powerdependence of Shapiro steps. In Fig. 5.12b we plot two linecuts from the datasetin (a). Dashed lines are spaced by 4(h/2e)fr. We identify equidistant Shapiroplateaus at VJ (∼ 2.75μV, see Eq. (2.28)) and 2VJ (∼ 5.5μV). In the top curve2VJ steps dominate because the width of the VJ plateaus is small at this particular

108


-7 -3.5 0 3.5 7I(nA)

919.15919.18919.21

f(M

Hz)

-0.2 0 0.2 0.4

V(μV)

I(nA)

-39dBm

-35

-68

P(d

Bm

)

919.15919.18919.21

f(M

Hz)

0 0.25 0.5

R(h/e2)

0 3.5 7

919.14MHz

-39dBm

(a)

(c)

-7 -3.5

919.17 919.18 919.19-2

0

2

4

f(MHz)V

(nV

)

Q=fr/Γ=520k

(b)

(d)

-63dBm,

I(nA)

f(M

Hz)-53

-39

Γ

ΔV∝ΔI

Figure 5.10: Shapiro steps on resonance with nanotube mechanics in deviceone. (a, top) Power dependence of Shapiro steps for a fixed frequency, f = 919.14MHz.In greyscale numerical dV/dI. (a, bottom) Shapiro steps as a function of RF frequency,for a fixed power, P = −39 dBm. (b) Zoom in on boxed part of (a). The black line isnot from this dataset, but is a scaled version of the linecut in panel (d). We added itto show that the shift (obviously) has similar lineshape as the mechanical resonance.(c) Dataset of panel (a,bottom), but now the measured voltage is plotted. Off resonantvoltage has been subtracted. (d) Linecut from similar dataset as in (c), but taken atlow power indicated by donut in (a). All data was taken at Vg = −8.715V.

(a) (b) (c)

-2

0

2

4

V(n

V)

919.17 919.18 919.19f(MHz)

Q=520k

-63dBm

0V(n

V)

f(MHz)

Q=110k

-39dBm

919.15 919.18 919.21-80

80

160

919.15 919.17 919.19-10

0

10

20

f(MHz)

V(n

V)

Q=340k

-53dBm

Figure 5.11: Mechanical resonator lineshapes measured at 7 nA in deviceone. Black line: measured data, grey line: Fano-function fit. (a) Lineshape measuredat −39 dBm (star in Fig. 5.10a). (b) Lineshape at −53 dBm (trapezoid in Fig. 5.10a).(c) At −63 dBm we find Q = 520k (donut in Fig. 5.10a, same dataset as in Fig. 5.10d).All data was taken at Vg = −8.715V, and I = 7nA.

power. Curves are shifted for clarity. Notice that the ∼ 8μV plateau at low poweris not due to phase locking of Josephson current to the applied microwave signal,but probably due to an electrical resonance in the bias circuit.

In device two we observe mechanical lineshapes similar to those observed in deviceone, shown in Fig. 5.12c-e. The quality factor of the resonator of device two isan order of magnitude smaller than that of device one. In (c) a small B field of46mT was applied, because the signal was small here at B = 0.

109


R(h/e2)

0 0.25 0.5 0.75

-7 -3.5 0 3.5 7-23

-68

I(nA)

P(d

Bm

)

I(nA)

V(

V)

1332.912MHz1332.912MHz

-2 -1 0 1 2-20

0

20

-35-68

1332.8 1332.95 1333.1-100

-50

0

50

f(MHz)

V(n

V)

Q=33k (B=46mT)

-7nA,-35dBm

1332.8 1332.95 1333.1-100

-50

0

50

f(MHz)

V(n

V)

Q=34k

7nA,-35dBm

1332.8 1332.95 1333.1-30

-15

0

15

f(MHz)

V(n

V)

Q=40k

7nA,-41dBm

(c) (d) (e)

(a) (b)

-35

Figure 5.12: Shapiro steps on resonance with nanotube mechanics in devicetwo. (a) Power dependence of Shapiro steps. (b) Two horizontal cuts of (a). (c-e)Lineshapes measured at the points indicated in (a) by a star, trapezoid and donut.

110

5.5. Mixing or rectification?

5.4.3 Summary and discussion of observed features

In three devices we observe a signal from the mechanical resonator as a currentshift. The signal is present in the power regime where Shapiro steps are. TheFano-lineshape and bias dependence are typical for mixing signals, which wasdiscussed in Sec. 2.5.1, and App. B.

In mixing experiments on CNTs two RF signals are present, usually one on theleads and the other on the gate. Fano-lineshapes are due to a phase differencebetween two mixing signals and have been repeatedly observed in devices thatare similar to ours [15–19]. The phase difference is set by the electromagneticenvironment of the device. As discussed in App. B, a Fano lineshape can alsooccur when only an oscillating gate voltage is present due to mixing of the gatevoltage with the nanotube motion, but we have estimated that this signal is verysmall for high Q devices such as studied here. The purpose of the followingsections is to clarify whether the observed signal is indeed due to mixing, and toinvestigate the role of Josephson dynamics and the presence of superconductivityin the leads.

5.5 Mixing or rectification?

DC currents such as we observe can always be mechanically generated in sus-pended CNTs, even in the absence of oscillating bias signals (see App. B). This

is due to terms with⟨y2⟩and

⟨Vgy

⟩in Eq. (B.9). The first term is referred to

as rectification while the latter is referred to as mixing. In addition, the RF an-tenna induces an oscillating gate voltage as well as an oscillating voltage/currentbias. The presence of both signals can result in an additional mixing signal (seeEq. (2.60)), that is also affected by mechanics. To distinguish between mixingand rectification we measure lineshapes as a function of gate voltage and currentbias.

Main resultIn both devices we find a regime where our signal is dominated by mixing insteadof rectification. This is indicated by the correlation to transconductance, and thebias dependence. We also find an increased signal in the SC regime close to I ∼ 0,that is not expected for normal mixing.

5.5.1 Dependence on transconductance

In the case of mixing we expect correlation between transconductance (∂G/∂Vg)and signal amplitude. For rectification, amplitude should be correlated to ∂2G/∂V 2

g .In our devices the transconductance can be changed with the gate voltage (Sec. 5.3).In Fig. 5.13 (device one) and 5.14 (device two), we present the gate voltage depen-dence. An extra method to distinguish mixing from rectification is by switchingbias polarity: Mixing signals are independent of bias polarity, while rectificationsignals change sign with bias polarity.

111


-100

916 918 920 922f(MHz)

0

200

V(n

V) -8.715V

7nA

-100

0

200

V(n

V)

-7nA

(c)

(a)

-50-50150

-9.75 -9 -8.25 -7.5-0.1

0

0.1

Vg(V)

-7nA

f-f r

(MH

z) V(nV)

(b)

-1000100

-0.1

0

0.1 7nA V(nV)

f-f r

(MH

z)

-9.75 -9 -8.25 -7.5 Vg(V)

-9.75VVg=-7.5V

Figure 5.13: Gate dependence device one. Gate dependence of the signal at±7 nA bias, at P = −39 dBm. Off-resonant voltage has been subtracted. The frequencyscan range was shifted to follow the resonance frequency fr, which was determined in aprevious measurement (data not shown). (a,b) Colorscale plots of the measured data.The blue dashed line indicates the Vg = −8.715V setpoint used in later measurements.(c) Linecuts taken from (a,b), labeled by the sweep frequency. Colored lines: Fit. Blacklines: Data.

112

5.5. Mixing or rectification?

02

-10

0

10-5.025V

1320 1325 1330 1335 1340 1345-10

0

10

Vg(V)

-7nA dV/df7nA dV/df

f-f r

(MH

z)

-6 -5.25 -4.5 -3.75Vg(V)

7nA

-7nA

-6VVg=-3.75V

-6 -5.25 -4.5 -3.75

-202

-0.45

0

0.45

f-f r

(MH

z)

-0.45

0

0.45

(c)

(a) (b)

f(MHz)

V(n

V)

V(n

V)

Figure 5.14: Gate dependence device two. Gate dependence of the signal at±7nA bias, at P = −35 dBm. Off-resonant voltage has been subtracted. The frequencyscan range was shifted to follow the resonance frequency fr, which was determined in aprevious measurement (data not shown). (a,b) Colorscale plots of the numerical deriva-tive dV/df of measured data. Blue dashed line indicates Vg = −5.025V setpoint usedin later measurements. (c) Linecuts taken from (a,b), labeled by the sweep frequency.Colored lines: fit, black lines: data. Important: We have stretched the lineshapes inthe horizontal direction by a factor of three, to enhance the visibility. This operation isshown in the inset in the top panel of (c).

113


In both devices we observe a gate dependence of the signal amplitude and achange of the signal sign. The resonance frequency increases almost linearly andthe quality factor does not show a strong dependence on gate voltage in bothdevices (data not shown). A change of signal sign as a function of bias polarity(I = ±7 nA) is not observed in either device. This indicates mixing signals arelarger than rectification signals.

5.5.2 Correlation to transconductance

We plot the maximum and minimum of fitted lineshapes in Fig. 5.15, and lookfor correlation to the transconductance.

-9.75 -9 -8.25 -7.5-0.4

0.4

Vg(V)

dG/d

Vg(

mS

/V)

0

-6 -5.25 -4.5 -3.75-0.8

0

0.8

Vg(V)

dG/d

Vg(

mS

/V)

I=-7nA

(b) (c)

-0.2

0.2

Vpe

aks(

V)

0

I=7nA-16

0

16

Vpe

aks(

nV)

-0.1

0

0.2

V(

V)

(a)

0

2

4

G(e

2 /h)

-9.75 -7.5Vg(V)

Vg=-8.715V Vg=-5.025V

fr

Figure 5.15: Correlation of signal to transconductance. (a) Illustration of whatis plotted in (b) and (c). Green dots in top panel show Vpeaks, the max/min voltage ofthe resonance lineshape. The transconductance is the derivative of G(Vg) to Vg. (b,top)Maximum/minimum of lineshapes as a function of gate, for I = ±7 nA. (b,bottom)Transconductance as a function of gate. (c) As in (b), but for device two.

We observe a strong correlation between the size of the signal and the size of thetransconductance across the full gate range (Fig. 5.15b). The size of the signaldoes not show a significant dependence on the bias polarity. This indicates thatin device one the signal is mostly due to mixing. In device two strong correlationis mostly absent (Fig. 5.15c), probably due to a competition between mixingand rectification. In this device we will measure at Vg = −5.025V, where themeasured signal appears to be dominated by mixing rather than rectification.

The sign of the signal is also correlated to the sign of the transconductance. Thisbecomes apparent after a comparison of Figs. 5.13 and 5.14 with the transconduc-tance gate dependence in Fig. 5.15. The sign can also be indirectly inferred fromthe top figures in panel 5.15(b,c). When the envelope is mostly positive/negative,the signal is pointing up/down.

We have now already established an understanding of the nature of our signal atB = 0, and conclude that it is mostly due to mixing.

114

5.6. Signal power dependence

5.5.3 Conclusion on mixing or rectification

We find gate regimes in both devices where bias and gate dependence show thesignal is dominated by mixing rather than rectification.

5.6 Signal power dependence

In this section we compare the dependence of the signal amplitude on drivingpower in the superconducting state to the normal state. We will quantify signalenhancement in the SC state. Also we use the power dependence to find out ifthe resonator amplitude can be detected in the linear regime, where y < yc (seeEq. (2.45)).

Main resultOur main finding is presented in Fig. 5.16. In the superconducting state, thesignal is up to two orders of magnitude larger compared to the N state. In deviceone for Vrf < 1mV the signal is only observed in the SC regime. A calibrationpoint at yc in device two allows for an estimation of the smallest amplitude thatcan be detected, which is at ∼ 3 pm on the order of twice the zero point motion.

0 1 2 3 4 50

150

300

450

Vrf(mV)

V(n

V)

2.4T

B=0T

0 1 2 30

400

800

Vrf(mV)

V(n

V)

B=0T

2.4T

(a) (b)

Device 1 Device 2

V

0

V

f

Figure 5.16: Signal amplitude as a function of driving voltage. Signal ΔV (Vrf),at B = 0T and B = 2.4T. (a) Data from device one. Inset shows how ΔV is extracted.(b) Data from device two, dotted grey lines are a quadratic fit to the low power partof the data, and a linear guide to the eye for high power data. Arrows indicate powersetting in the field dependence data (Sec. 5.7).

115


To quantify the transduction of applied driving power to signal amplitude ΔVwe zoom in on the regime where ΔV = 0 . . . 50 nV (Figs. 5.17-5.21). The deviceshave different power dependence and we will discuss them separately.

5.6.1 Power dependence

Technical detailsAll data has been measured at I = 7nA. We have scaled the applied powerP (dBm) to a driving voltage by V ∝

√P (W ). For example −30 (dBm) → 1mV.

The actual RF voltage across the device is determined by the impedance of thesetup and is not known.

Device oneIn device one we measure only linear ΔV (Vrf) dependence (Figs. 5.17-5.18).Such dependence is expected when the non-linear resonator is driven above yc(Fig. 2.18). We observe that in the N regime (at B = 0T) we have to apply twoorders of magnitude larger Vrf to acquire a signal with a similar amplitude as inthe SC regime.

With Eq. (2.46) we estimate Fc ≈ 1.4 · 10−18 N, using E = 1.25TPa, r = 1.4 nm,L = 210 nm, and Q = 3 · 105. With y2c ≈ r2/Q we extract yc ≈ 2.6 pm. As wedo not observe the crossover from the linear to the quadratic regime we concludethat our sensitivity is not large enough to detect amplitudes y < 2.6 pm, and thatthe driving force F > 1.4 · 10−18 N.

14 29 440

0.3

0.6

Vrf( V)

Q(1

06)

14 29 440

25

50

Vrf( V)

V(n

V)

(a) (b)

919.14 919.165 919.19-15

0

15

30

f(MHz)

V(n

V)

(c)

Figure 5.17: Low power regime, device one, SC contacts. (a) ΔV (Vrf) atB = 0T. Grey lines: linear fits, slope ΔV/ΔVrf ≈ 1.5 · 10−3. (b) Quality factor.(c) Lineshapes taken at points indicated by diamonds in (a,b). Grey lines: Fano fits.The dashed horizontal line indicates the noise floor. A single data line was acquired byintegration of 800 frequency sweeps of 150 points. Data acquisition was done at 4.4Hz(1.5ms/pt), the total integration time per line was 3 minutes.

116


1 3 50

0.15

0.3

Vrf(mV)

Q(1

06)

918.98 919.02 919.06

0

30

f(MHz)

V(n

V)

1 3 50

25

50

Vrf(mV)

V(n

V)

(a) (b) (c)

Figure 5.18: Low power regime, device one, normal contacts. (a) ΔV (Vrf) atB = 2.4T. Grey line: linear fit, slope ΔV/ΔVrf ≈ 13 · 10−6. (b) Quality factor. (c)Lineshapes taken at points indicated by diamonds in (a,b). Grey lines: Fano fits.

Device twoIn device two we can fit the superconducting response ΔV (Vrf) at B = 0T, verywell with a quadratic function ∝ V 2

rf (Fig. 5.19a). We notice that there appearsto be a crossover from a quadratic to linear dependence. The crossover pointis indicated by the vertical line at Vrf = 0.3mV. Such behavior is expected formixing detection with a Duffing resonator (see Fig. 2.19c). The crossover point isat the critical force Fc. We can use this point as a calibration point to estimatethe size of the smallest amplitude we can detect. We also notice that becausedevice two has a lower Q-factor than device one, its critical driving amplitude ycis larger than that in device two. This makes it in principle easier to measurey < yc in device two, which is consistent with our observations.

With Eq. (2.46) we estimate Fc ≈ 4 · 10−18 N, using E = 1.25TPa, r = 1.4 nm,L = 360 nm and Q = 0.5 · 105. With y2c ≈ r2/Q we extract yc ≈ 6 pm. We candetect signals down to ≈ 0.5Fc, which implies that the smallest amplitude wecan measure in device two is ymin ≈ 3 pm. This value should be compared to theamplitude of thermal fluctuations:

kBT/2 = m(2πfr)2y2th/2 , (5.1)

and the amplitude of zero point motion fluctuations [20]:

yzpm =√

(�/(4πmfr)). (5.2)

We find yth ∼ 2.4 pm at T = 90mK, and yzpm ≈ 1.5 pm.

To our knowledge, such sensitivity is unprecedented in CNT resonators. Withoutsuperconductivity we are not able to detect such small signals. We have to applyF > Fc to detect a signal when superconductivity is suppressed by magneticfield. The agreement of ymin ≈ yth suggest that the amplitude of the resonatoris driven mainly by thermal motion, and that our sensitivity is large enough todetect it. One way to check this would be to do a power dependence measurementat elevated temperature, and observe a saturation of amplitude at yth > ymin. InFig. 5.19b we have plotted lineshapes measured at the diamond points indicatedin Fig. 5.19a. We notice that the two curves taken at high power do not fit verywell to the Fano-lineshape, which indicates non-linear behavior. At low powerthe Fano-lineshape fits better, which is more clearly visible in Fig. 5.20c. This isanother indication for the crossover between linear and non-linear behavior.

117


0 0.5 10

175

350

Vrf(mV)

V(n

V)

(a) (b)

1332.7 1332.9 1333.1-200

-100

0

100

f(MHz)V

(nV

)

F Fc

Figure 5.19: Linear/non-linear crossover in device two. (a) ΔV (Vrf) at B = 0.Grey line: quadratic fit to low-power data. Vertical line is at Vrf = 0.3mV. (b)Lineshapes taken at points indicated by diamonds in (a). Grey lines: Fano fits.

In Figs. 5.19-5.21 we present the small signal part of Fig. 5.16b. In the SC stateat B = 0, we have an enhancement of the signal by at least a factor of fourcompared to the N state at B = 2.4T (here we assume that the amplitude ofthe resonator does not increase in this power range beyond Fc). In other words,in the N state we have to apply four times larger driving voltage to acquire thesame signal size as in the SC state. The linear dependence on Vrf at B = 2.4Tis consistent with driving above Fc.

We notice that the resonance linewidth is effected by the magnetic field. AtB = 0T, Q ∼ 50k, and at B = 2.4T, Q ∼ 30k.

0.50 0.250

20

40

60

0 0.25 0.50

0.5

1

Vrf(mV)

Q(1

05)

Vrf(mV)

V(n

V)

(a) (b)

1332.8 1332.95 1333.1-50

-25

0

25

f(MHz)

V(n

V)

(c)

Figure 5.20: Low power regime, device two, SC contacts. (a) ΔV (Vrf) atB = 0T. Grey line: quadratic fit. (b) Quality factor. (c,d) Lineshapes taken at pointsindicated by diamonds in (a,b). Grey lines: Fano fits. A single data line was acquiredby integration of 200 frequency sweeps of 150 points. Data acquisition was done at4.4Hz (1.5ms/pt), the total integration time per line was 45 s.

118


1332.7 1332.95 1333.2-25

0

25

50

f(MHz)

V(n

V)

0.5 1 1.50

30

60

Vrf(mV)

V(n

V)

0.5 1 1.50

0.5

1

Vrf(mV)

Q(1

05)

(a) (b) (c)

Figure 5.21: Low power regime, device one, normal contacts. (a) ΔV (Vrf) atB = 2.4T. Grey line: linear fit. (b) Quality factor. (c) Lineshapes taken at pointsindicated by diamonds in (a,b). Grey lines: Fano fits.

5.6.2 Possible origin of large signal in superconducting state

It is reasonable to assume that the Coulomb driving force is proportional to Vrf .Also, when the resonator is driven beyond yc the amplitude of the resonator doesnot increase. Therefore we have to conclude that the increase of signal amplitudeis probably due to enhanced mixing in the SC state.

In App. A we discuss our experimentally determined order of magnitude estima-tion for transduction enhancement in the SC regime compared to the N regime. Inthe SC regime we find that the signal is about two orders of magnitude enhancedin device one, and about one order of magnitude in device two. In Eq. (2.67) wehave presented a theoretical order of magnitude estimation for Josephson mixingrelative to normal mixing. In our experiments Josephson mixing would appearas an enhanced resonance signal in the superconducting regime. We observe suchenhancement, and if it is due to Josephson mixing that would imply that wehave underestimated Im,J/Im,N by one or two orders of magnitude. This is notunthinkable, since (as we have mentioned before) the “real” critical current couldbe about one order of magnitude larger than the switching current.

An alternative explanation is transduction enhancement by a change of impedancein the SC vs. N state, that results in larger mixing currents in the SC state. Un-fortunately we have no experimental means of obtaining the impedance. In theremaining sections we will compare the N regime to the SC regime, by studyingthe dependence of the signal on magnetic field and temperature to learn more onthe origin of the signal.

5.6.3 Conclusion on power dependence

Sensitivity to driving voltage is in the SC state up to two orders of magnitudelarger compared to the N state. This allows us to estimate the smallest amplitudewe can detect, which is on the order of: y ∼ 3 pm, only twice the zero pointmotion.

119


5.7 Magnetic field dependence

In this section we present our data on the magnetic field dependence of theresonance signal. The magnetic field is directed along the trench, in-plane withthe film.

Main resultOur main finding is that the signal vanishes almost completely above Bc ∼ 2.1T,which is the critical field of the leads. We also find a surprising sign change. Ata particular field the resonance lineshape vanishes, and reappears flipped upsidedown. This field is B ≈ 0.3T in device one, and B ≈ 1.9T in device two. Wehave a strong indication that the signal is due to Josephson mixing.

This also indicates that our device is a Zeeman π-junction, and that the sign-change measurement of the resonance lineshape is due to a reversal of the su-percurrent direction. This also indicates that we perform a phase-sensitive mea-surement of the supercurrent, because the reversal of supercurrent direction isreflected in the sign change of the mechanical response.

5.7.1 Magnetic field dependence in device one

In Fig. 5.22a we plot lineshapes as a function of field in a colorplot. We find thatthe signal is symmetric in bias, which indicates mixing. Above Bc the amplitudeof the signal is below the noise floor (at the power used). We have measured upto B = 4.5T to emphasize the contrast between the normal and superconductingstate. Also, at B = 3 . . . 4.5T a weak resonance signal is present that is barelyvisible. The change of sign occurs at B0−π,1 = 0.3T. In Fig. 5.22b we plotthe minimum and maximum value of the peaks, which shows the decay andreappearance of the signal.

(a) (b)

B(T)0 0.7 1.4 2.1

-50050

V(nV)

0 1.5 3 4.5B(T)

-50050

V(nV)

I=-7nA

-100

0

100

V(n

V)

0.3T

I=7nA

919.14

919.18

919.22

f(M

Hz) I=-7nA

919.14

919.18

919.22

f(M

Hz) I=7nA

Figure 5.22: Field dependence, device one. (a) Colorplot of measured voltage.Off resonant voltage has been subtracted. (b) Signal minimum and maximum amplitudeas a function of field. Data has been taken at Vg = −8.715V and P = −44 dBm.

Because the presentation of data as in Fig. 5.13c, (plot lineshapes next to eachother) allows for an easy way to display the signal amplitude and sign, we haveplotted our data in a similar way here. Linecuts have been plotted next to each

120

5.7. Magnetic field dependence

other, with x-axis labels that indicate the magnetic field at which the linecutsare taken. To avoid confusion we illustrate this procedure explicitly in Fig. 5.23.

0mT 60mT 120mT

0V(n

V)

f(MHz) f(MHz) f(MHz) 0 60 120B(mT)

Figure 5.23: Shifted-linecut plots. Three linecuts have been taken at B = 0mT,60mT and 120mT. We plot them next to each other in a new panel, where they arelabeled by the bias point at which they were taken. Grey lines: fits, black lines: data.

In Fig. 5.24a we have plotted shifted linecuts from the dataset shown in Fig. 5.22.Linecuts taken in the vicinity of B0−π,1 are presented in 5.24b. The sign changeis apparent.

-100

100

V(n

V)

0

0 0.7 1.4 2.1B(T)

-20

0

20

V(n

V)

919.15 919.2-30

0

30

f(MHz)

V(n

V)

I=7nA

I=-7nA

0.32T0.51T

0.18T0.41T

I=7nA

I=-7nA

(a) (b)

-100

100

V(n

V)

0

Figure 5.24: Resonance lineshapes as a function of field, device one. (a) Wehave plotted linecuts from the datasets in Fig. 5.22a next to each other. (b) Linecutstaken in the vicinity of B = 0.3T. Data has been taken at Vg = −8.715V and P =−44 dBm.

5.7.2 Magnetic field dependence in device two

In device two we find similar results as in device one. The signal disappears(mostly) above Bc, and there is a change of sign at B0−π,2 = 1.94T. Thisdataset is shown in Fig. 5.25. In Fig. 5.26d we have plotted linecuts taken fromthis dataset in the vicinity of B0−π,2, making the sign change clearly visible.A changing sign could be due to a change of sign in transconductance, or thelocal resistance dV/dI. To check if this is the case, we have measured the fielddependence of resistance and transconductance, which we discuss next.

121


(c) (d)

-100

0

1332.87

1333

1333.13f(

MH

z)V(nV)

-100

0

V(nV)I=-7nA

I=7nA

0

20V(nV)

100

01.88 1.94 2.00

B(T)

V(nV)

(a) (b)

-30

0

30

V(n

V)

I=-7nAI=7nA

0 1.5 3 4.5B(T)

-120

-60

0

60

V(n

V)

I=-7nAI=7nA

B(T)

1332.6

1332.9

1333.2

f(M

Hz)

1332.6

1332.9

1333.2

f(M

Hz)

1332.87

1333

1333.13

f(M

Hz)

1.88 1.94 2.00B(T)

0 1.5 3 4.5

I=-7nA

I=7nA

1.94T

Figure 5.25: Field dependence, device two. (a) Colorplot of signal as a functionof field. Off resonant voltage has been subtracted. The sign change occurs in theregion inside the dashed boxes. (c) Zoom-in on the dashed box area in (a). (b) Signalamplitude as a function of field. (d) Zoom-in on the dashed box area in (c). Data hasbeen taken at Vg = −5.025V and P = −35 dBm.

-40

0

40

V(n

V)

-20

0

20

V(n

V)

1MHz

1.91 1.94 1.97B(T)

1MHz

I=7nA

I=-7nA

Figure 5.26: Resonance lineshapes as a function of field, device two. Shiftedlinecuts as a function of field, taken from datasets in Fig. 5.25c. Data has been takenat Vg = −5.025V and P = −35 dBm.

122


5.7.3 Detailed analysis of the field dependence signal

When the magnetic field is changed, also the conductance of the device changes,due to the effect of the magnetic field on the states of the quantum dot (Sec. 2.1.1).We have done our experiments at a gate voltage where the transconductancedG/dVg does not change much in field, to minimize effects on the mechanicalsignal. This allows us to extract a signal that is proportional to the mechanicaldisplacement.

To isolate field dependence of mixing with y from field dependence of dG/dVg,the signal has to be corrected for the “gain” by transconductance. Also the localdV/dI close to the bias point (I = ±7 nA) changes as a function of field, andshould be taken into account. We first calculate the mixing current ΔIm thatgave rise to ΔV , using the local dV/dI and then divide by the transconductancedG/dVg. This gives a signal that is proportional to the amplitude Δy, Δy =κΔIm/(dG/dVg). Here κ = 1/2(V0/C0)(∂C/∂y) sin(ϕ), where sin(ϕ) = 0 in casethe AC Josephson effect is suppressed, and sin(ϕ) ≈ 1 otherwise. We present ourdata on device one in Fig. 5.27.

0 0.9 1.8 2.40

0.25

0.5

B(T)

0

125

250

B(T)

0

60

120

B(T)

0

0.1

0.2

B(T)

0

10

20

0 0.9 1.8

0 0.9 1.80

0.5

1

dG/d

Vg(

S/V

)dV

/dI(

h/e2

)

V(n

V)

y(

AV

/S)

(a) (b)

(c) (d)

Im (pA)

Isw (nA)

Bc=2.1T

0 0.9 1.8

Figure 5.27: Overview of resistance, transconductance and signal amplitudeas a function of field, device one. (a) Resistance dV/dI as a function of field, inunits of h/e2 = 25.8 kΩ. At Bc = 2.1T a small jump of ∼ 1.5 kΩ indicates the criticalfield of the contacts. (b) Signal size ΔV . Mixing current ΔIm is defined as ΔV/(dV/dI).(c) Transconductance as a function of field. Transconductance is reduced by a factorof two between B = 0 and B = 0.9T. (d) By dividing the mixing current by thetransconductance we extract a signal Δy ·κ that is proportional to the CNT amplitude.Data taken at Vg = −8.715V and I = 7nA.

In panel (a) we plot the resistance as a function of field. Our device is connectedsuch that a short part of the superconducting leads is always in series with thenanotube Josephson junction (Fig. 5.2b). We contribute the small jump in resis-

123


tance at B = 2.1T to the critical field of the leads. In panel (b) we plot ΔV andΔIm = ΔV/(dV/dI). The resistance does not change much as a function of fieldin the range of B = 0 . . . 2.4T. The transconductance changes considerably, thisis shown in panel (c). This change is due to a change of the quantum dot stateas a function of field (by mechanisms discussed in Sec. 2.1.1). The origin of thetransconductance change is not our main interest. The important observation isthat it does not change sign in this field range.

In panel (d) we show the signal proportional to amplitude, Δy ·κ. At B0−π thereis a cusp. We find that Δy · κ grows with further increase of field. We haveplotted the field dependence of Isw as well. In Sec. 5.7.8 this will be discussed indetail, but we have included it here to get an idea of the relative field scales onwhich Isw and ΔV decay. Similar data on device two is presented in Sec. A.2.3.

5.7.4 Possible origin of increased signal at B < Bc

For the same reasons as discussed in Sec. 5.6 we can attribute the enhanced signalat B < Bc to the change of device impedance with field, or to an extra contri-bution to mixing signals by Josephson mixing. A “simple” change of impedanceonly, is not sufficient to explain our data, because it has to account for a signchange as well. We will show next how Josephson mixing can account for a signchange of mixing signal.

5.7.5 Possible origin of signal sign change at B0−π

In case the large mixing signal originates from Josephson mixing, we model it (tofirst order) in the following way (see also Sec. 2.5.2):

Im =1

2

∂Ic∂Vg

V0

C0

∂C

∂y〈y sin (ϕ)〉 . (5.3)

In Sec. 5.3.2 we have shown that in our devices the switching current is correlatedto the conductance of our device. This implies that the ∂G/∂Vg ∝ ∂Ic/∂Vg andthat the sign of mixing signals due to Josephson mixing is correlated to thetransconductance as well.

We have measured the field dependence of the transconductance and find thatit does not change sign (see Fig. 5.27(c,g)). In our model this implies that thefactor 〈y sin (ϕ)〉 has to change sign.

One way in which this could occur is when sin (ϕ) → sin (ϕ+ π) = − sin (ϕ),which would give:

Im,π =1

2

∂Ic∂Vg

V0

C0

∂C

∂y〈y sin (ϕ+ π)〉 = (5.4)

Im,π = −1

2

∂Ic∂Vg

V0

C0

∂C

∂y〈y sin (ϕ)〉 . (5.5)

Such phase shift of π can be picked up when the ground state of the Josephsonjunction is changed from ϕ = 0 to ϕ = π, which can be achieved with a magnetic

124


field (see Sec. 2.6.3). We notice that we find B0−π in both devices has the rightorder of magnitude. As we will show in the next section, we have found that itis also tunable with gate. Our simple model does not account for this.

Field dependence of resonance frequency and quality factorThe fits also yields the field dependence of resonance frequency and quality factor.This data is presented in Sec. A.2.6. Further experiments need to be done to gainunderstanding of this data.

5.7.6 Gate voltage dependence of B0−π

We have measured the gate voltage dependence of B0−π in device one, and wefind that it is gate tunable between B0−π ∼ 0.15 . . . 0.9T. In Fig. 5.28a,c wepresent our data on the gate dependence of B0−π.

0

0.45

0.9

B (

T)

1 2 3 4

fr,1 f

-9.75 -9 -8.25 -7.50

0.45

0.9

B0-

(T)

Vg(V)

B0-

-0.4

0.4

dG/dV

g (mS

/V)

0

1 2 3 4

B0-

fr,2 f fr,3 f fr,4 f

0.3

0.4

0.5Isw/Isw,max

-9.75 -9 -8.25 -7.50

0.45

B (

T)

Vg(V)

B0- 1 2 3 4

0 0.15 0.450

0.5

1

B(T)0.3

I sw/I s

w,m

ax

1234

(a) (b)

(c)

(d)

Vg

Figure 5.28: Gate and field dependence of mechanical resonance and su-percurrent, device one. (a) Four datasets taken at the gate voltage 1-4 indicatedby the vertical dashed lines in panel (c). In colorscale is the measured voltage, similarto the dataset in Fig. 5.22a. (b) Normalized switching current as a function of field,taken at the gate voltage 1-4. (c) Extracted B0−π from 51 datasets like those presentedin panel (a). In case there was no sign change or signal, there is no datapoint. (d)In colorscale: Normalized switching current. Overlay: Dataset from panel (c). Gatevoltages 1-4 correspond to: Vg,1−4 = −8.085,−8.535,−8.76,−9.12V. Bias current wasI = 7nA and applied power in panel (a,c) was P = −38.66 dBm. Width of panels in(a): 2Δf = 45 kHz.

We have taken similar datasets as presented in Fig. 5.22a, but now as a function ofgate voltage. Four example datasets in which the tunability of B0−π is apparentare presented in panel (a). From 51 of such datasets we have extracted B0−π(Vg),which is shown in panel (c). We find that the dependence has mostly an oscillatingnature, with the exception of three datapoints where B0−π > 0.45T. With theaim to learn more about the mechanism of B0−π, we have compared its gatedependence to G(Vg), Isw(Vg) and IswRn(Vg), but did not find any correlation.

125


We do however find a resemblance in the gate dependence of the transconductance,which is also shown in panel (c) (we used the same dataset as in Fig. 5.15b).

We have also measured the gate dependence of the (normalized) switching cur-rent: Isw(B), to look for a correlation between switching current and B0−π. Aswill be discussed in Sec. 5.7.8, a 0−π transition can effect Isw(B). When the tran-sition is tunable with gate voltage, this can perhaps show up in Isw(B). We do ob-serve a weak gate dependence of the foot-height (Isw/Isw,max(0.4 > B > 0.25T)).Four linecuts are shown in panel (b), and the full dataset is shown in colorscalein panel (d). The weak gate dependence is apparent in the colorscale, but is notclearly correlated to B0−π(Vg) (overlay in panel (d)).

In Sec. 2.5.1 we found that the size of the mixing current is proportional to thetransconductance. When we neglect the three datapoints at B0−π > 0.45T, ourdata suggests there is an average B0−π ∼ 0.25T that is tunable by ±0.12T withthe amplitude of the mixing current.

5.7.7 Signal bias dependence as a function of magnetic field

In this section we present bias-dependence data as a function of magnetic field.We are particularly interested in comparing data measured when the leads are inthe superconducting/normal state. We present this data here because we wantto investigate if the magnetic field induced sign change depends on the currentbias.

In Fig. 5.29-5.30 we present data on bias dependence. We have used the shifted-linecut plot again.

Bias dependence device oneAt B = 0 the sign of the signal does not change across the whole bias range(indicating mixing) and it is pointing up. At B = 0.3T the signal amplitude issuppressed, and we notice a sign-change indicating rectification at |I| ∼ 0, butmixing again at |I| � 0. This is indicated by the blue fits in Fig. 5.29c. Our datasuggests that at this field mixing is suppressed at |I| ∼ 0 more than at |I| � 0.At B = 0.6T the signal amplitude is revived and we notice again that the signdoes not change, but now it is pointing down.

We find that the sign change due to magnetic field persists across the whole biasrange, but not at B = B0−π, which indicates that mixing signals disappear atB = B0−π, while rectification signals do not.

126


(a)

B=0

B=0.31T

B=0.61T

(c)

-1500

150300

V(n

V)

-50

0

50

V(n

V)

-150

050

V(n

V)

-70 -35 0 35 70I(nA)

919.12

919.15

919.18

f(MH

z)

V(nV)

0

200

-70 0 70I(nA)

(b)

0

200

400

V(n

V)

-70 0 70I(nA)

V

919.12 919.18f(MHz)

0T

0.31T0.61T

Figure 5.29: Bias dependence in SC and N state, device one. (a) Colorplotof measured voltage as a function of frequency and bias at B = 0. Off-resonant voltagehas been subtracted. (b) Signal amplitude ΔV as a function of bias and field. (c)Vertical linecuts from dataset in (a), and linecuts taken from similar datasets measuredat B = 0.31T and 0.61T (colorplots not shown). Colored lines: fits, black lines: data.Blue lines show rectification. Data taken at Vg = −8.715V and P = −39 dBm.

127


Bias dependence device twoAt B = 0 the data looks similar to that of device one, the sign of the signal doesnot change as a function of bias. At B = 1.8T we find that the sign of the signalchanges at high bias, indicative of a transition from mixing to rectification. AtB = 2.4T (this is above Bc, as we will discuss in Sec. 5.7), the signal sign haschanged with respect to B = 0. At I ± 70 nA the signal is mainly due to recti-fication, indicated by the asymmetric signal polarity. Detailed field dependencewill be discussed in Sec. 5.7. In this device we also observe an asymmetry insignal amplitude at large bias, which is possibly the result of the bias-dependedcompetition between mixing and rectification.

B=0

B=1.8T

B=2.4T

f(MH

z)

-80-400

V(nV)

1331.2

1331.5

1331.8

-90 0 90

-100

0

0

50

-25

0

25

V(n

V)

V(n

V)

V(n

V)

-70 -35 0 35 70

2.4T 1.8T

0T

0

75

150

V(n

V)

-90 0 90

1331.2 1331.8f(MHz)

I(nA) I(nA)

I(nA)

(a)

(c)

(b)

Figure 5.30: Bias dependence in SC and N state, device two. (a) Colorplotof measured voltage as a function of frequency and bias at B = 0. Off-resonant voltagehas been subtracted. (b) Signal amplitude ΔV as a function of bias and field. (c)Vertical linecuts from dataset in (a), and linecuts taken from similar datasets measuredat B = 1.8T and 2.4T (colorplots not shown). Colored lines: fits, black lines: data.Data taken at Vg = −5.025V and P = −35 dBm.

Apart from the sign change, the most striking feature we find in both devicesis an increased mixing signal close to I = 0. The increase of signal cannot beexplained by non-linearity of dV/dI as a function of bias (see technical detailsbelow).

This bias dependence is not expected for regular mixing/rectification, where thesignal increases with the DC voltage across the device (see Eqs. (2.60) and (2.63)).The increased signal is only present at B = 0, which suggest that mixing closeto I = 0 is enhanced by superconductivity. However, it is not directly clearwhy it would drop as a function of bias due to suppression of AC Josephsoncurrents, because there is no bias dependence of Ic in the RCSJ model. This

128


means that (within the RCSJ model) the current spectral density does not changeas a function of bias. If mixing is really enhanced by superconductivity due tomixing with AC Josephson currents the bias dependence should be explained bya model that involves suppression of AC Josephson current away from I = 0.

Technical detailsThe resistance of the device also changes as a function of bias and field. In App. Awe present the measured bias dependence as presented in the above figures, andthe bias dependence of dV/dI. We use it to extract the mixing current that givesrise to the voltage peak. We find that the nonlinearity of dV/dI is not sufficientto explain the increased signal in the SC state. The dip in signal amplitude atI = 0 and B = 0 in device two (Fig. 5.30b) is caused by a small dV/dI. In deviceone we have taken more datapoints and the variation of dV/dI on Shapiro stepsis apparent as an oscillating ΔV (I) in Fig. 5.29b. In following experiments wewill bias our devices at I = ±7 nA.

5.7.8 Magnetic field dependence of supercurrent

In this section we present the field dependence of switching current and Shapirosteps. Our motivation for these experiments is to look for evidence of π-junctionbehavior. In Fig. 2.24 we present typical behavior of Ic in the 0 − π transitionregime, nodes appear at the 0 − π crossing. We have looked for such nodes inour data, by measuring Isw as a function of field.

Main resultIn both devices switching currents disappear in field. In device one this happensat B ∼ 0.5T, and in device two at B ∼ 1T. Curves have a foot-like structurewith a pronounced kink. We do not find clear cusps in our data, only in devicetwo there is a small “bump”-like feature. The position of kink/bump is notsignificantly tunable by gate, also the number of Shapiro steps is constant as afunction of field.

Field dependence of switching currentIn Fig. 5.31 the switching current appears as a sharp jump in dV/dI (white linesin colorscale). Below this line the junction is in the SC state. For device one,at B > 0.5T, Isw is not well defined. We notice two similar features in bothdevices: A small increase of Isw at low field, and a kinked, foot-like structure atlarger fields. At large field there is not a clear switching current anymore, but isreplaced by a series of small, vanishing steps.

We find the kink position in device one at B = 0.23T. In device two Isw isnot well defined anymore at B > 1T. The kink of the foot structure is here atB ∼ 0.5T, and we observe a tiny bump in Isw(B) at B ∼ 0.76T. This bump isindicated by the white arrow.

129


0

0.25

0.5

B(T)

I(nA

)

Device 1(a)

0

0.6

1.2

0

0.5

1

0 0.5 1 1.50

0.8

1.6

B(T)

I(nA

)

Device 2

0 0.25 0.5

(b)

dV/dI (normalized/line) dV/dI (normalized/line)

Figure 5.31: Switching current as a function of field. (a)/(b) Device one/two:dV/dI as a function of field and current bias. The colorscale has been stretched foreach line. The dataset in (a) has been taken at Vg = −8.715V, the dataset in (b) atVg = −5.15V.

130


Gate and field dependence of switching current and Shapiro stepsIn Figs. 5.32 and 5.33 we present the gate dependence of Isw(B) and the fielddependence of Shapiro steps. Our motivation is to see if the foot-structure istunable by gate, and we will look for a field dependence of Shapiro steps. As wasdiscussed in Figs. 2.26 and 2.27, in case the sin(2ϕ) term of the current phaserelation is present, this can result in fractional Shapiro steps. In our experimentwe foresee that the current phase relation can be tuned by magnetic field, whichcould be probed by Shapiro steps. Here we will investigate if such features appearin our device.

Device oneIn Fig. 5.32 we show our date for device one. In panels (a,c) we observe thatIsw(B) does not depend on Vg. In panel (b) we have plotted two linecuts takenfrom (a,c). In panel (d) we plot a histogram of measured IV curves under irra-diation of the junction at f = 2GHz. Constant voltage Shapiro plateaus appearas lines in the histogram. Integer Shapiro steps at hf/(2e) = 4μV are dominant,but decay in strength as the field is increased. Fractional steps appear in betweenthe integer steps and become stronger when the field is increased from B = 0 toB = 0.3T.

-9.75

-9

-8.25

-7.5

B (T)B (T)

V g(V

)

B (T)

V g(V

)

-6.5

0

6.5 V(V)

Vg=-8.715V,f=2GHz

B (T)

(a) (b)

0.6

1.2

0

1

3

5

-16

-8

04 V

-4

0

4

dIsw /dB(nA

/T)

0

0.6

1.2

I sw(n

A)

0 0.25 0.5

0 0.25 0.5

0 0.25 0.5

0 0.25 0.5

8

-9.75

-9

-8.25

-7.5

Isw(nA)

Vhist(counts)

dIsw/dB (nA/T)

(c) (d)

Figure 5.32: Switching current and Shapiro steps as a function of field,device one. (a) Switching current as a function of gate and field. (b) Linecuts fromthe datasets in (a,c), taken at Vg = −8.175V, indicated by dashed horizontal lines in(a,c). The grey curve is the derivative of the black curve. (c) Here we plot dIsw/dB, tolook for gate dependence of the kink-position. The kink-position does not change in thisgate range. (d) Histogram of IV s measured at Vg = −8.175V with 2GHz irradiation.

131


Device twoIn Fig. 5.33 we show our data for device two. It is mostly similar as in device one,but here we find a small bump in Isw(B). This is mostly apparent in Fig. 5.31,and not very visible here, because it is a small bump. In panel (c) it is justobservable in the colorplot. The bump position is indicated by the arrow, anddoes not shift with gate voltage.

In this device we observe broad fractional Shapiro steps already at B = 0, andthey have similar field dependence as the integer Shapiro steps. This is shown inpanels (b,d).

-3.75

-4.5

-5.25

-6

B (T)B (T)

Vg(

V)

B (T)

Vg(

V)

V(μ

V)

B (T)

(a) (b)

V(μ

V)

0 0.5 1

0 0.5 1

0 0.5 1

0 0.5 1

Isw(nA)

Vhist(counts)

dIsw/dB(nA/T)

(c) (d)

-3.75

-4.5

-5.25

-6

00.511.5

0.511.5

-3.3

0

3.3

0.511.5

Vg=-5.15V,f=3GHz

Vg=-4.43V,f=3GHz

-6

0

6

-6

0

6

Vhist(counts)

Figure 5.33: Switching current and Shapiro steps as a function of field,device two. (a) Switching current as a function of gate and field. (c) dIsw/dB,to look for gate dependence of the kink and bump position. (b,d) Histogram of IV smeasured when the junction was irradiated with a microwave signal at 3GHz. Constantvoltage Shapiro plateaus appear as lines in the histogram. Integer Shapiro steps at 6μVare dominant, but decay in strength as the field is increased. Broad fractional stepsappear in between the integer steps. The dataset in (b) is taken at Vg = −5.15V and in(d) at Vg = −4.43V. These gate voltages are indicated by the dashed horizontal linesin (a,c).

Enhancement of Isw at low magnetic fieldWe have noticed a peculiar increase of Isw at B = 0 . . . 25mT in device one, andB = 0 . . . 120mT in device two. Such behavior has also been observed in thinsuperconducting MoGe wires (at 2 . . . 4T), and Au/Nb squids (at∼ 2mT) and Znnanowires (at ∼ 3mT) [11,21,22]. One explanation is given by the disappearanceof Cooper pair-breaking due to spin-flip scattering on magnetic impurities, whenmagnetic moments are polarized at a magnetic field of μBBp = kBT [21]. Thisprocess can be important when the Zeeman energy of electrons is larger than thethermal energy, which is ∼ 65mT in our experiments. At T = 90mK we find

132


Bp ∼ 130mT, which is similar to the value observed in device two. We concludethat we have a possible explanation for the enhancement of Isw at low magneticfield. Other mechanisms that are described in Ref. [21] can perhaps also play arole here.

5.7.9 Conclusion on magnetic field dependence

First we will discuss the main features we have observed in the signal magneticfield dependence, and then we will present our conclusions.

Foot structure in Isw(B)In Fig. 2.24 we have shown a typical dependence of Isw(B), that has also beenmeasured in diffusive junctions [23,24]. Our observed foot structure deviates fromthe typical curves, most prominently: It has no node. It is reasonable to assumethat the current phase relation I(ϕ) in our devices is non-sinusoidal, because weare in the T � 1 limit (see Fig. 2.11). In this regime also sin(2ϕ) componentscan have sizable contribution to the supercurrent. This is also supported by thepresence of fractional Shapiro steps in both devices, already at B = 0.

In Sec. 2.6.3 we have given an example of a possible field dependence of criticalcurrent in ballistic π-junctions. We note here that our data is consistent with thesimple model sketched in Fig. 2.26, notably we find that the sign-change field atwhich sin(ϕ) changes sign, can be at a field that is above the field where the kinkin the foot structure appears.

The absence of gate dependence on Isw(B) indicates that if the foot structure isdue to π-junction physics, it does not depend strongly on the conductance. Wenote that in our ballistic π-junction model presented in Fig. 2.25, the B0−π fielddepends on the induced proximity gap Δ. In a follow up experiment one couldtry to measure the dependence of B0−π on Δ, by for example a temperaturedependence measurement of B0−π.

We conclude that it is in principle possible to explain our observations with a0 − π-transition and a non-sinusoidal current phase relationship, but that theswitching current and Shapiro step field dependence are not typical signatures ofregular 0− π-junction behavior.

Sign change in mechanical responseIn device one the sign change occurs at B0−π = 0.3T, which corresponds roughlywith the position of the kink in Isw(B) observed in Fig. 5.32. In device two thereis a larger discrepancy, the kink position is at B = 0.5T, and the sign changeat B = 1.9T. In device one we have found that B0−π can be tuned with gatevoltage, and the majority of datapoints are correlated to the transconductance.This indicates that B0−π can be modulated with the amplitude of the mixingcurrent.

Because we cannot think of any other mechanism than Josephson mixing thatwould give a sign change in the response, we conclude that we have a strongindication that the signal is due to Josephson mixing. This also suggests thatour device is a Zeeman π-junction. It indicates that our single Josephson junc-

133


tion devices allow for a phase-sensitive measurement of the supercurrent (seeSec. 2.6.4), because the change of supercurrent direction is reflected in the signchange of the mechanical response.

Bias and field dependenceIn both devices we find an enhanced mixing signal at I ≈ 0. Regular mixing orrectification does not have such behavior. As the enhancement is only presentat B = 0, this suggest that mixing is enhanced by superconductivity. in deviceone, we find at B = B0−π a rectification signal at I ≈ 0. This suggests that therelative amount of mixing and rectification can be tuned by magnetic field. Weconclude that bias dependence also indicates Josephson mixing.

As a control measurement we have measured the temperature dependence. If ourassumption of Josephson mixing is correct, the signal has to disappear at T > Tc.

5.8 Temperature dependence

The superconductivity in the contacts is expected to disappear above Tc ∼ 2.4K,the critical temperature found when we characterized our rhenium films (seeApp. D). In this section we will present our data on the temperature dependenceof the signal.

Main resultWe find that the signal almost completely disappears at T > Tc.

Technical detailsThe temperature of the device is increased by switching on a heater on the mixingchamber of the dilution fridge. We start with very small heating power and try tolet the temperature increase slowly in the range of 500mK to 2.5K. During thistime we measure the signal and the mixing chamber temperature. The resistanceof the device can be extracted from the (off-resonance) measured voltage. Thisallows us to plot the signal ΔV and the resistance R vs. the temperature T .It should be noted that the measured temperature is only an indication of thereal temperature, because the mixing chamber resistance has a weak temperaturedependence above T = 2K. Also the device temperature can lag behind on themixing chamber temperature due to the finite heat capacity of the fridge partsbetween the temperature sensor and the device.

5.8.1 Temperature dependence in device one

In Fig. 5.34a we observe a strong decay of the signal at T = 2.2K. In Fig. 5.34cwe present ΔV (T ) and R(T ). We observe that the resistance increases gradually,while the signal first decays and then drops at T = 2.2K. The resistance increasesby 2 kΩ, which we attribute to the resistance of the Re contacts.

134

5.8. Temperature dependence

In Fig. 5.34c we have also plotted Q(T ). We observe a weak decay of Q withincreasing temperature. In the temperature region indicated by the grey box thesignal disappears. In Fig. 5.34b we show linecuts taken at temperatures aboveand below Tc. The weak decay is consistent with the high-temperature region ofthe Q ∝ T−0.36 power law [20,25].

(a) (b)

0.5 1.5 2.5024

T(K)

Q(1

05)

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60

120

V(n

V)

6.5

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)

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Hz)

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0 30 60 900.5

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)

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80

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V(n

V)

T=0.5K

T=0.8K

T=1.5K

T=2.6K

V

(c)

f(MHz)919.13 919.16 919.19

t(min)

Figure 5.34: Signal amplitude as a function of temperature, device one. (a)Measured lineshapes as a function of time. Off-resonant voltage has been subtracted.Black curve: measured temperature. (b) Linecuts taken from (a). Numbers indi-cate temperature and Q-factor, curves are offset for clarity. (c,top) Red curve: Peakheight ΔV extracted from fits to the data in (a). Green curve: Measured resistance.(c,bottom) Q-factor extracted from linewidth of the fits. Grey area indicates T > Tc.All data was taken at Vg = −8.715V, P = −44 dBm and I = 7nA.

5.8.2 Temperature dependence in device two

In device two we observe similar features as in device one. Our data is presentedin Fig. 5.35a. We observe a strong decay of the signal, that is accompanied bya steep increase of the device resistance by ∼ 2 kΩ. In this experiment we haveextracted the device resistance from IV curves taken in a small range aroundI = 7nA.

135


In Fig. 5.35b we present ΔV (T ) and R(T ). We observe that the resistanceincreases steeply close to T = 2K, while the signal first decays and then jumpsdown close to T = 2K. The resistance changes by ∼ 2 kΩ, which we attributeto the resistance of the rhenium contact lines to the CNT. In the bottom panelwe have plotted the extracted Q-factor. In Fig. 5.35b,d we show linecuts takenat temperatures above and below Tc. Notice the y-axis scale is one order ofmagnitude smaller in the normal state.

1330.875

1331.375

1331.875

f(M

Hz)

0 80 1607.5

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)

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(k)

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1331 1331.4 1331.8-80

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80T=2K

T=360mK

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1331.3 1331.7 1332.1

-8

0

8

f(MHz):

T=2.41K

T=2.36K

V(n

V)

V

V(n

V)

f(MHz)

(d)

Figure 5.35: Signal amplitude as a function of temperature, device two.(a) Measured voltage as a function of time. Off-resonant voltage has been subtracted.Green curve: Measured resistance. (c,top) In red: Peak height ΔV extracted fromfits to the data in (a). Green curve: Measured resistance. (c,bottom) The Q-factorextracted from linewidth of the fits. Grey area indicates T > Tc. (b,d) Linecuts fromdataset in (a). Curves are offset for clarity. All data was taken at Vg = −5.025V,P = −35 dBm, and I = 7nA.

5.8.3 Conclusion on temperature dependence

We conclude that there is a strong correlation between the signal magnitude andthe presence of superconductivity in the leads, which is consistent with Josephsonmixing.

136

5.9. Mechanical resonance at Shapiro plateaus

5.9 Mechanical resonance at Shapiro plateaus

5.9.1 Introduction

MotivationUntil now we have only discussed resonance signals that were taken at a currentbias of ∼ 7Isw. This is mainly for convenience, since here dV/dI is more orless constant as a function of other parameters. Oscillating Josephson currentsare still expected to be present at our bias point, but we do not exactly knowhow their amplitude depends on applied power or bias. At low power and lowbias (Shapiro step regime), we can calculate the power spectral density of theJosephson current, as we have presented in Fig. 2.16.

The presence of such oscillating currents in our devices is apparent in the Bessel-like power dependence of the Shapiro steps. But we also notice that our Shapirosteps are not sharp, but smoothed (Figs. 5.10 and 5.12). Such broadening couldbe due to thermal noise (see Sec. 2.3.3), but we do not know the exact mecha-nism. We can probably assume that the Josephson linewidth is broad and thespectrum of Fig. 2.16 is smeared out, making individual current componentsbarely distinguishable.

Nevertheless there could be weak features of the spectrum surviving in our de-vices. If for example a Bessel-type power-dependence as in Fig. 2.16 would showup, this would immediately be a smoking-gun signal for Josephson mixing. Theother smoking-gun signal, Josephson force driving, will occur when there is a res-onance condition between mechanical vibrations and Josephson oscillations. Thiscan only happen at Shapiro plateaus. This motivates us to measure the mechan-ical resonance signal in the Shapiro step regime. We have done this experimentonly in device one.

Main resultWe find in one dataset a non-trivial power dependence, the signal increases whilethe power is decreasing. This suggests that the driving force and/or the mixingcurrents have a non-trivial power dependence, which is expected for Josephsonforce driving and/or mixing with Josephson currents.

5.9.2 Method of measurement

The current can only be inferred indirectly from the measured voltage. Thisis done numerically by division of the measured voltage by the local resistance,which we also measure by taking an IV in a small range at each bias pointwith the same S/N ratio. At low power, we observe that the local resistancevaries by four orders of magnitude. This makes extracting the current a verytedious business. At low power the signal is very small and frequency sweepshave to be integrated for a long time to yield a sufficient signal to noise ratio. Anextra complication is that we cannot measure ΔI at very flat Josephson plateausbecause there ΔV ∼ 0. At such points we have no data.

137


5.9.3 Measurement results

The results of this experiment have been summarized in Fig. 5.36. We willcarefully describe each panel. Our goal is to measure the amplitude of mixingcurrent ΔIm as a function of power, at two bias points.

Panel (a): Where the experiment is done.In panel (a) we show the part of the Shapiro-step pattern in which we do theexperiment. This is the low power regime. The dominant red and blue plateausare the first two Shapiro plateaus. The bias asymmetry is due to hysteresis ofthe junction. We will measure the mechanical resonance as a function of power,at two different bias points indicated by the two dashed green lines. The starindicates a reference point that will reappear in other panels. This panel servesas a reference to make clear where the experiment is done.

Panel (b): The mapping procedure.In panel (b) we show the mapping procedure. Remember that our goal is toextract the amplitude of mixing signal ΔIm. To achieve this we tune to the powerand bias point where we will extract ΔIm. We measure first an IV curve at aslightly off-resonant excitation frequency (V (I)). Notice that this has alreadybeen done in panel (a), but that we need a much higher resolution and S/Nratio than we have there, because ΔV is very small. Secondly we measure theresonator lineshape in the usual way by sweeping the resonance frequency throughmechanical resonance.

Then we use a numerical mapping technique to map V (f) to I(f) via V (I).Mapping is done as follows:

� Interpolate the measured IV curve such that the spacing between data-points is 1 nV.

� Round the data in V (f) to values that are integer multiples of 1 nV.

� Take values of the trace V (f), and look up to which current I(f) thiscorresponds on the IV .

In this way we can extract I(f) even when dV/dI is not constant in the range ofΔV = |max(V (f))−min(V (f))|. Example traces are shown in panel (b). Noticethat the vertical scale, V , holds for V (f) and V (I), and that the horizontal scale,I, holds for I(f) and V (I). The labels ΔV and ΔI indicate the signal amplitude.

Every extracted curve was evaluated and only kept if there was an apparentresonance peak, and a good fit to the Fano lineshape. Curves that were not goodhave not been used in panel (c).

Panel (c): The extracted data.In this panel we show the amplitude of the mixing signal ΔIm. These are thegreen squares in the plot. In the top panel we show the data taken at I = 0.23 nA.The data in the bottom panel is taken at I = 0.

The colorplot in the background is a histogram of IV traces such as in panel (b).This makes it immediately clear if the bias/power set point is on a flat plateau

138

5.9. Mechanical resonance at Shapiro plateaus

(sharp peak in histogram) or on a step in between two plateaus (histogram withbroad peaks).

In the top panel we observe two cusps in the power dependence of the signal andthe position of the main “bump” appears to be correlated to a high voltage bias.In the lower panel we do not observe a significant power dependence.

Panel (d): Extracted data vs. ΔV/ΔI.We have linearized V (I) in the range ΔV = |max(V (f))−min(V (f))|, by divid-ing through ΔV/ΔI, to estimate the local slope. We have marked ΔIm valuesabove the noise floor by circles, and values that were below the noise floor bycrosses. These values measure the noise in the mapped I(f) trace rather thanΔIm. We now see that there is a cutoff at ΔV/ΔI ≈ 1 kΩ, below which thesignal disappears. There is no apparent correlation between the local slope andthe mixing signal amplitude.

-25 0 251.80

1.85

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)

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V(

V)

I=0.23nA

(c)

-64 -53P(dBm)

data for one datapoint in (c)

Figure 5.36: Mechanical resonance at Shapiro steps. (a) Shapiro step patternwith lines indicating the range where mechanical resonance will be measured. (b)Mapping procedure to extract current (green line) from measured signal (red line), andmeasured IV (blue line). Star indicates bias point. (c) Green squares: Extracted data.Colorplot: Voltage histogram of local IV . (d) ΔV/ΔI is the local resistance wherethe resonance peak was measured. Here we plot all extracted peak heights ΔI thatwere above the noise floor (circles), and below the noise floor (crosses) as a function ofΔV/ΔI. Crosses indicate a peak to peak noise value of mapped current in I(f) traceswhere no resonance signal was observed.

139


Technical detailsThe effect of Josephson dynamics on Shapiro steps is highly dependent on theapplied power. In Fig. 5.10b we observe that the width of the Josephson plateauis at high power (P = −39 dBm) comparable to the width of the step connectingtwo plateaus. Furthermore, the plateau is not very flat, but has a slope byitself. Such behavior is typical for circuits with non-linear impedance. Largeoscillating signals are rectified, which causes DC signals. Effectively this appearsas smoothed Shapiro steps. For very high power the steps disappear.

At low applied power (P = −53 dBm) plateaus are very flat and the steps con-necting them are very steep. According to Eq. (2.27), the Josephson current hasa narrow bandwidth when the bias voltage is stable. For example, when the volt-age is 2.068μV ± 20 nV the corresponding Josephson current has a frequency of1GHz±10MHz. This corresponds to QJc = 100 (we use Jc to indicate Josephsoncurrent). A sloping Josephson plateau indicates a broad spectral density of thecurrent going through the junction at this plateau.

By approximation the Josephson linewidth is given by the vertical width (ΔV )of a plateau. For the frequency range of interest (∼ 1GHz) we find at high powerΔV ∼ 1μV → 500MHz. At very low power the plateaus are flat and in thisregime we expect a narrow linewidth Josephson current, but there we cannotextract mixing currents.

5.9.4 Conclusion on mechanical resonance at Shapiro steps

We have investigated the power dependence of the mixing signal as a functionof power and bias in the low power regime. In this regime the power spectraldensity of the Josephson current can depend strongly on voltage bias and appliedpower. In a resonant situation Josephson force can drive the motion of the CNT,and its amplitude should also oscillate with applied power.

In our measurements we do observe one “oscillation” of the mixing signal as afunction of power, which could indicate an oscillating power dependence of PSD,or very weak Josephson driving. These two mechanism cannot be distinguishedin our data. Our data is consistent with mixing and/or driving by Josephsoneffect, we do not know how normal mixing and gate force driving can give rise toincreasing signals when the driving power is lowered.

140

5.10. Observed features and general conclusions

5.10 Observed features and general conclusions

In this section we summarize our observations and provide conclusions.

5.10.1 Observed features

We list the notable features of device characterization, the nature of the measuredsignal at B = 0, and compare the magnitude of the signal in the SC and the Nstate.

Device characterization

� We observe the Josephson effect and Fabry-Perot interference in clean sus-pended CNTs with GHz mechanical modes.

� Maximum conductance of our devices is ∼ 4e2/h.

Nature of the measured signal at B = 0

� The signal has mostly a Fano-type lineshape.

� The size and sign of the signal is correlated to the transconductance.

� The sign of the signal does not change in a large bias range.

Superconducting state vs. normal state

� Sensitivity to driving voltage is in the SC state up to two orders of magni-tude larger compared to the N state.

� In the SC state the signal size doubles close to zero bias.

� Bias dependence is easily suppressed in magnetic field and almost absentin the N state.

� Below Tc the signal is at most one order of magnitude larger compared toabove Tc.

� The signal drops sharply at T ∼ Tc.

� The signal drops by at most two orders of magnitude at fields above Bc,‖.

� In the SC state the signal changes sign as a function of field.

� The field at which the sign change takes place is tunable by gate, and ismostly correlated to the transconductance.

Measured signal on Shapiro plateaus

� At Josephson plateaus we observe an oscillating mixing current as a functionof power.

141


5.10.2 Conclusions

In our experiments at high bias the nanotube motion can be explained with gateforce only. At low bias, on Shapiro plateaus, we measure a power dependencethat cannot be explained with a gate force and normal mixing. Mixing signalsare much larger in the SC regime compared to the N regime. We conclude thatthe signal is most likely due to mixing with Josephson currents. We come to thisconclusion based on the following arguments.

Arguments for signal originating from mixing

� Gate and bias dependence of mechanical signal point to mixing rather thanrectification.

� Observed Fano-lineshapes are another indication of mixing.

� The dependence on driving voltage is consistent with mixing.

Arguments for signal originating from Josephson mixing

� Temperature and field dependence shows that the signal is enhanced byup to two orders of magnitude in the presence of superconductivity in theleads.

� The sign-change as a function of field is in agreement with a simple modelfor Josephson mixing and Zeeman splitting of ABS in a ballistic junction.

� The order of magnitude of the enhanced signal in the SC state is compati-ble with an estimation of the ratio between Josephson mixing and normalmixing.

� The power dependence at Shapiro plateaus is inconsistent with gate forcedriving and mixing with normal current.

Argument against Josephson mixingWe have considered the possibility that the larger signal in the SC state is dueto a change of circuit impedance when the leads become superconducting. Wehave not determined the impedance of the whole circuit and the effect of super-conductivity, and hence we cannot predict how signals are affected. It is notimmediately clear to us how a change of impedance can result in a sign changeof the resonance signal.

Phase-sensitive measurement on CNT Zeeman π-junction

� Josephson mixing allows us to do a phase-sensitive measurement on thesupercurrent.

� By magnetic field, the junctions can be tuned from groundstate at ϕ = 0to ϕ = π.

Enhanced sensitivity due to Josephson mixing

� Josephson mixing enhances the sensitivity to mechanical vibrations, whichallows us to detect the CNT amplitude down to ∼ 3 pm, which is on theorder of only twice the zero point motion, and comparable to the expectedthermal motion of the resonator.

142

5.10. Observed features and general conclusions

5.10.3 Context and recommendations

Theoretical predictionIn parallel with the experiment we have worked on a theoretical description ofthe effect of mechanical resonance on Josephson dynamics, which is included inApp. C. We predict a large force that depends on the superconducting phasedifference and could drive the mechanical resonator in the presence of resonantsupercurrents. This Josephson force has a power dependence that is very dif-ferent from the gate force. Whereas the gate force increases linearly with thedriving voltage Vrf , the Josephson force oscillates and is maximum at low Vrf (seeFig. 2.22). In our theoretical work we have assumed that the CNT motion isstrongly driven by the Josephson force, but we have not experimentally observedthis.

Josephson force in our devicesWe have looked for signatures of the Bessel-function dependence of Joseph-son force and Josephson mixing on driving voltage Vrf . On low-power Shapiroplateaus this is not easy because there the signal is very small. The (voltage)signal magnitude is dominantly determined by the local slope of IV curves, whichcan change by four orders of magnitude on/off Shapiro plateaus. In one devicewe find an oscillating power dependence. Such oscillations could be explainedwith Josephson force driving, or an oscillating power dependence of Josephsoncurrent that is on resonance with mechanics. Our experiment does not allow usto distinguish between these two. The signal is very weak (long integration timesare necessary), while we expected it to be very large.

We suggest that the large (or total) suppression of Josephson force is due tothe absence of strong resonance between the Josephson force and mechanicalvibrations. This would only occur at bias voltages where ωr = ω0 = 2eV0/�,with ωr the resonator frequency and ω0 the Josephson frequency. The linewidthof Josephson signals and Josephson force is probably very broad due to thermalnoise. This reduces the efficiency of energy transfer to the mechanical resonator,which has a very narrow linewidth. We suggest that this is the main reason forthe absence of strong Josephson driving in our data; There is no efficient couplingof Josephson force to CNT motion. Efficiency could be enhanced by reducing thelinewidth of Josephson signals.

A broad linewidth of Josephson currents also results in a small DC mixing prod-uct. In Fig. 5.37 we give a sketch of the linewidth mismatch and the effect onthe mixing products.

Towards Josephson force driving and mechanically induced ShapirostepsWe suggest that to observe driving of high Q CNT resonators by Josephson force,and to achieve larger mixing signals (even in the absence of Josephson driving) itis necessary to stabilize the bias voltage down to the level of the CNT resonatorlinewidth. This corresponds to a stability of 0.1 nV, if we consider a 1GHzresonator with Q = 20k. Shunting the junction with a low impedance wouldreduce the thermal noise, but it is not clear what has to be done to sufficientlystabilize the voltage.

143


-1 0 1 2 30

f(fr)

PS

D(a

.u.)

mechanicsJosephson

currentmixingproduct

Figure 5.37: Reduced mixing product due to linewidth mismatch. While thelinewidth of mechanical resonance is very sharp, the linewidth of Josephson currents isprobably very broad. This results in a small mixing product.

144

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146

Chapter 6

Mechanical resonance at afractional driving frequency

In this chapter we discuss an experiment performed on device one (see Fig. 5.5),where we have probed the mechanical resonance by driving the system at a frac-tion of the resonance frequency. Our goal is to investigate the possibility of para-metric driving the resonator, and detection of the mechanical resonance usingthe AC Josephson effect.

6.1 Parametric excitation and detection by Joseph-son mixing

Due to the nonlinear nature of the Josephson Eqs. (2.26) and (2.27), a Josephsonjunction driven by an RF signal with frequency f , will also carry currents atfractional and integer multiples of f . In Sec. 2.3.3 we have presented the analysisof the current spectral density of a driven Josephson junction as a function ofvoltage and power. In this chapter we will study the frequency dependence ofour signal. We will look for signals when we drive the system off-resonance atinteger fractions of the resonance frequency.

Typically a CNT resonator is driven at ωrf = 2πfCNT, where fCNT is the res-onance frequency of the CNT and ωrf the applied microwave frequency. Thismicrowave signal capacitively induces oscillating signals in the CNT device, thatare all oscillating at the same frequency. In a typical mixing experiment thegate force (see Eq. (2.50)) drives the CNT motion, and the combination of CNTdisplacement with oscillating gate and bias voltages can result in mixing or rec-tification signals (see App. B), that can be measured at DC. This situation isshown in Fig. 6.1a. We note that in this configuration the RF signal that is usedto detect the CNT motion is directly driving the motion as well.

In the case of directly driven motion, the excitation force is on resonance with amode of the resonator at ω0. In the case of parametric excitation, the motion isdriven by a force that is a fraction of ω0, it is done by driving one of the parameters

147

6. Mechanical resonance at a fractional driving frequency

in the equation of motion at a frequency ωp =2

nω0. In CNT resonators this is

done by modulating the spring constant [1]. In this way extra energy can bepumped in the resonator, which amplifies motion at ω0. This is similar to thechild on a swing, that increases its amplitude of motion by periodically shiftingits center of mass. Parametric excitation has been recently studied on a CNTresonator [2].

In principle the AC Josephson effect allows for a way to detect parametricaldriven motion, by mixing down mechanical signals to DC using harmonics of thedriving signal that are generated by the AC Josephson effect.

When we excite our device at the fractions 1/2ω0 and 1/3ω0, possibly its motionwill be driven by parametric excitation. In the case of normal contacts, up tofirst order there will be no oscillating bias or gate voltages induced that are onresonance with the CNT motion, and for this reason the usual mixing techniquesare less efficient to probe parametric excited motion. We note that when higherorder terms are included in Eq. (2.57) we can also account for higher-order currentterms that can mix down parametric excited motion. Such terms are for exampleproportional to V 4

sd, which makes them small. Nevertheless parametric excitedmotion has been detected in this way [3].

The situation is different when the CNT is a Josephson junction, because therethe current spectrum contains sizable integer multiples of the driving frequency,that can (potentially) mix parametric excited motion down to a DC signal. Thisis shown in Fig. 6.1, which is a simplified version of Fig. 2.16.

0 g

rf

Sig

nal p

ower

3 J2 JJCNT

g0

rf

biasCNT

Sig

nal p

ower

(a) (b)

Figure 6.1: Detection scheme using a fractional driving frequency. (a) ACNT with normal contacts is driven on resonance to find a DC mixing signal. Up tofirst order all driving forces are on resonance with the CNT. (b) When the CNT formsa Josephson junction, the current spectral density contains sizable harmonics at integermultiples of ωrf , that potentially can be used to probe the mechanical resonance. The“direct” driving forces are now off resonance with the CNT. Its motion can be excitedparametric (not shown in this figure).

In our experiment we will try to find a resonance signal when the system is drivenat a subharmonic resonance frequency. In case superconductivity is present weexpect to find a signal, under the condition that the motion of the CNT is drivensufficiently strong, by parametric excitation or thermal motion.

148


6.2 Characterization

In this section we present characterization of device one (see Fig. 5.5), which wasdone for the experiment described in this chapter. Here we operate in a differentregion of gate space compared to Ch. 5. In Fig. 6.2a we show two gate traces. Theblack line is the gate trace that was taken just before we started the experimenthere, and the grey line is the gate trace that we have taken from Fig. 5.5. Wenote that they do not overlap exactly and discuss this in the technical detailsbelow.

The most striking feature in the gate trace are the sharp jumps that occur atVg < −12V. We believe that these jumps are probably due to charging of a non-conducting second nanotube (CNT 2) that is in the vicinity of the conductingnanotube (CNT 1). When the gate voltage is Vg < −12V, the charge on aclosed quantum dot (in the Coulomb blockade regime) in CNT 2 is changing withdiscrete steps. Because the CNTs are capacitively coupled, the change of chargeon CNT 2 causes a change of the induced charge on CNT 1, which effectivelyshifts the gate trace. CNT 1 acts as a charge sensor to CNT 2. We have noindication that current is actually flowing through CNT 2. Similar data has beenreported on a similar device with two CNTs before [4].

When we measured the jumping gate trace for the first time, our interpretationwas that CNT 2 must be a very large bandgap CNT, such that the Schottkybarriers are very large and the current flowing through CNT 2 is to small todetect. The geometry of the device was expected to be similar to that shown inFig. 3.13. To our surprise we found a second CNT when we inspected our devicein SEM, but instead of crossing the trench it was suspended between one contactand the dielectric. This is shown in Fig. 6.2b,c. Our hypothesis of charge sensingis consistent with magnetic field dependence of the jumps, that will be discussedin Fig. 6.7.

We started our experiments on device one with the gate voltage tuned on a sharpjump, because there the mechanical resonance signal was large and easy to detect(this will be discussed in Sec. 6.3.2). Because this conductance feature is atypicalwe shifted our operation point in between jumps, at a position in gate space thatis representative for a typical CNT device. This is shown in Fig. 6.3. We willdiscuss our results in this gate range in Sec. 6.3.1, and the results in the gaterange across a jump in Sec. 6.3.2.

We observe a tunable mechanical resonance of fr ∼ 960MHz, and a gate dependedsupercurrent that is correlated to the conductance.

Technical detailsHistorically, this experiment was done before the experiments described in Ch. 5.In between both measurement runs there were two thermal cycles (due to a failureof the dilution fridge) in which in one case the device was warmed up to roomtemperature, and in the other case it was warmed up to 4K. We believe thatthis is the main reason that the characteristics are different from those presentedin Fig. 5.5.

149


Vg(V)-18 -12 -6

0

2

4

G(e

2 /h)

(a)

(b) (c)

CNT 2

CNT 1

CNT 2

qCNT2=1e

Figure 6.2: Gate trace and SEM images. (a) Gate traces taken on device one.The black trace is taken just before the experiment discussed here. The grey trace istaken from Fig. 5.5 (see text). Sharp jumps are probably due to charge sensing of asecond CNT. (b) SEM image showing CNT 1 and CNT 2. (c) Zoom in on CNT 2.

150


0

max

955

960

965

970

f(M

Hz)

dV/df(a.u.)

0

G(e

2 /h)

-18 -17.5 -17 -16.5Vg(V)

2

4

G(e

2 /h)

Vg(mV)

2.5

3

3.5

I sw(n

A)

0

0.5

1

-40 400

(a) (b)

-1 0 1-10

010

I(nA)V

(V

)Isw

Figure 6.3: Electrical characterization. (a, top) Mechanical resonance as a func-tion of gate voltage. We plot dV/df to enhance contrast. The inset shows the gaterange (panel (b,bottom)) in which part of the experiment will be done. (a, bottom)Conductance as a function of gate voltage. The box indicates the gate range used fordata in panel (b). Dashed lines indicate two gate settings that were used in followingdatasets. (b,top) Switching current Isw as a function of gate voltage ΔVg. Gate volt-age is indicated as an offset to Vg = −17.163V (black dashed line in panel (a,bottom)).The inset shows a typical IV . (b,bottom) Conductance as a function of gate voltage(zoom in from box in panel (a,bottom)).

151


6.3 Experimental results

We have done our experiments in two separate gate ranges, in the first gate rangethe gate dependence of the conductance is typical (away from the jump). Thisregime will be discussed in Sec. 6.3.1. The second gate range is in the vicinityof a jump, where the conductance shows atypical behavior. This regime will bediscussed in Sec. 6.3.2.

6.3.1 Experimental results away from sharp charge transi-tion

The experiment is relatively straightforward. We first find a mechanical modeof the resonator. This is shown in Fig. 6.4a,d. Then we drive the system atan integer fraction of the mechanical resonance frequency fr. We have usedfrf = 1/2fr and frf = 1/3fr. In Fig. 6.4b,e and 6.4c,f we show our measureddata. We find a signal when driving the system at these fractions of fr. Thelower panels (d,e,f) show linecuts taken from (a,b,c). We use them to extract thefit parameters quality factor (see caption) and resonance frequency.

We find fr = 961.4471MHz, fr,1/2 = 480.7247MHz and fr,1/3 = 320.4823MHz.This gives the fractions fr/fr,1/2 = 2.0000 and fr/fr,1/3 = 3.0000. We find asignal when driving at a precise integer fraction of the mechanical resonancefrequency. We also note here that we are not simply driving higher harmonics ofa resonator mode, because fr,1/2 is not an integer multiple of fr,1/3.

Higher harmonics are typically not at precise integer multiples of the fundamentalmode, because the spatial distribution (the shape) of the mode causes a differentelectronic environment that changes the resonance frequency (this is called slack).When a mode is excited parametrical, its shape is not different compared toresonance excitation, and as a consequence the resonance frequency is found at

the precise integer fractions2

nω0 [3, 5].

Caveat: driving with spurious harmonics from RF sourceWhen interpreting our results we have to be very careful, because we cannotcompletely exclude the possibility that we are driving the device with higherharmonics that are generated by the RF source itself. The specifications of theRF source we have used (Rohde & Schwarz SMB100A) list an attenuation of the2f harmonic of < −30 dBc and of the 3f harmonic of < −40 dBc in the frequencyrange we used [6]. This implies that we could be driving the system at a higherharmonic of the carrier frequency, with a power of ∼ −70 dBm. This shouldbe compared to the smallest power we have used to find mechanical resonance.This is shown in Fig. 5.10d, where with a power of −63 dBm we got a signal ofΔV ∼ 5 nV. Although that measurement was done at a different frequency andfor this reason we cannot compare them directly, we find that it is surprising thatour signal has the same order of magnitude while the applied power in the secondand third harmonic is one order of magnitude smaller.

152


(a) (b) (c)

961.1 320.4

320.6

min

0

max

-40 0 40Vg(mV)

dV/df

f (M

Hz)

0

480.5

480.9 dV/df

min

max

f (M

Hz)

-40 0 40Vg(mV)

961.7

0

0.4

f (M

Hz)

V(mV)

-40 0 40Vg(mV)

961.147 961.447 961.747-0.4

0

0.4

f(MHz)

V(

V)

480.710 480.725 480.740-10

0

10

20

f(MHz)

V(n

V)

320.467 320.482 320.497

0

20

f(MHz)

V(n

V)

(d) (e) (f)

Figure 6.4: Signal at fractional driving frequency. All data was taken at I ∼0 nA, tuned on a Shapiro plateau (large dV/dI). (a) Data taken at P = −58.2 dBm. (b)Data taken at P = −43 dBm. (c) Data taken at P = −30 dBm. (d) fr = 961.4471MHz,Q = 66.5k. (e) fr,1/2 = 480.7247MHz, Q = 265k. (f) fr,1/3 = 320.4823MHz, Q =225k. Linecuts are taken at Vg = −17.171V, indicated by the dashed vertical lines. Offresonant voltage has been subtracted.

153


We have measured a signal by fractional driving the system at 1/2fr with P =−58 dBm and at 1/3fr with P = −36 dBm. This is shown in Fig. 6.5.

480.715 480.730 480.745-10

0

10

20

f(MHz)

V(n

V)

320.477 320.487 320.497-10

0

10

20

f(MHz)

V(n

V)

Q=298k Q=354k

P=-58dBm (P2f<-88dBm) P=-36dBm (P3f<-76dBm)

(a) (b)

Figure 6.5: Low power signal at fractional driving frequency. (a) P =−58 dBm, fr,1/2 = 480.7301MHz, Q = 298k, I ∼ 0 nA. (b) P = −36 dBm,fr,1/3 = 320.4874MHz, Q = 354k, I ∼ 0 nA. Off resonant voltage has been subtracted.

At such low powers the harmonic at 2f is at < −88 dBm and at 3f it is < 76 dBm.The observed signal is the same order of magnitude as that found in Fig. 5.10d,while the spurious components at 2f and 3f are now at a power that is roughlytwo orders of magnitude below that used in Fig. 5.10d. This makes it unlikelythat we are detecting motion driven by spurious components of the RF source,and we find that our hypothesis of parametric excitation, and detection by mixingdown mechanical motion with Josephson currents is more likely.

Magnetic field dependence of mechanical resonance signalIn the measured magnetic field dependence at fractional driving we find that thenature of the signal is similar to that found in Ch. 5. This is shown in Fig. 6.6.At this gate setting: Vg = −16.86V, we find B0−π ∼ 555mT. Our data is similarto that observed in Fig. 5.28, and is consistent with Josephson mixing.

Bc

959.37 319.80

319.82 V(nV)

f (M

Hz)

479.70

479.73V( V)

f (M

Hz)

959.48

f (M

Hz)

V( V)

0 0.8 2.4B(T)

(a) (b) (c)

B0-

1.6 0 0.8 2.4B(T)

1.6 0 0.8 2.4B(T)

1.60

0.1

0.2

-20

0

20

-0.2

0

0.2

Figure 6.6: Field dependence of signal at fractional driving. We measuredthe mechanical resonance as a function of field, driving the system on resonance andat fractions of the mechanical resonance. We find B0−π = 555mT. All datasets havebeen taken at Vg = −16.86V. (a) Data taken at P = −61 dBm. (b) Data takenat P = −34 dBm. (c) Data taken at P = −30 dBm. Off resonant voltage has beensubtracted.

In the next paragraph we present the magnetic field dependence of the conduc-tance, to confirm that it is not the transconductance that is changing sign asfunction of field.

154


Magnetic field dependence of conductanceIn Fig. 6.7 we plot the field dependence of the conductance. In panel (a) wefind that the sharp jumps move in field. The field dependence of the jumps isan indication that they are due to charge sensing of charge transitions in CNT2. Such shifting and kinking of lines is possible when there is a B‖ componentto CNT 2 (theory: see Sec. 2.1.1, experiment: [7–9]). The orientation of themagnetic field is in-plane with the contacts, along the direction of the trench.From the SEM image in Fig. 6.2c, it is clear that there is a field componentparallel to CNT 2.

In panel (b) we plot a linecut taken along the dashed line in panel (a). We observea gradual decrease of the conductance with field, and a sharp jump at Bc = 2.1T,that we identify as the critical field of the contacts. In panel (c) we plot dG/dVg,to make sure the transconductance does not change sign as a function of field.This dataset is similar to that presented in Fig. 5.27.

Vg(

V)

G(e2/h)

0 0.8 2.4B(T)

(a)

1.62

3

4

-17.2

-16.9

-16.6

Bc=2.1T

2.4

2.8

3.2

G(e

2 /h)

0

6

12

dG/d

Vg(

e2/h

V)

0 0.8 2.4B(T)

1.6 0 0.8 2.4B(T)

1.6

(b) (c)

Figure 6.7: Field dependence of conductance. (a) Colorscale plot of conductanceversus field and gate voltage. The dashed line indicated the position of linecuts. Noticethe conductance jumps kink with field. (b) The linecut shows a gradual decay of theconductance, and a sharp decay at Bc = 2.1T, the critical field of the contacts. (c) Thederivative of the conductance with respect to the gate voltage does not change sign.

Magnetic field dependence of switching currentIn Fig. 6.8 we plot normalized dV/dI as a function of field. This plot is similarto Fig. 5.31, the switching current Isw appears as bright datapoints in this plot.We find that the foot structure is also present at this gate voltage, but that it isless pronounced compared to Fig. 5.31.

155


0.6 1.20

0.6

1.2

B(T)

I(nA

)

00

0.5

1

dV/dI (normalized/line)

B0-

Figure 6.8: Switching current as a function of field. We plot dV/dI as a functionof field and current bias. The colorscale has been stretched for each line. The datasethas been taken at Vg = −16.93V.

156


6.3.2 Experimental results on sharp charge transition

In this section we will present our data on the mechanical resonance when tuningthe gate voltage through the sharp conductance jump that we attribute to chargesensing of a charge transition in CNT 2. We originally started our experimentson device one in this gate range, because (for a given power) the mechanicalresonance signal is roughly three orders of magnitude larger when the gate voltageis tuned on such a sharp transition, compared to a typical gate voltage regime.We will first present the device characterization in this regime and point out thedifference in signal amplitude. Next we present our results of fractional drivingin this regime, and show that we can measure the response of the mechanicalresonator when driving the system at a quarter of its resonance frequency.

CharacterizationIn Fig. 6.9 we show the characterization of our device in this gate regime. Inpanel (a) we show the gate dependence of the supercurrent branch. We find thatit makes sharp jumps as well. This gate dependence cannot be directly mappedto the gate traces presented in Fig. 6.2, because the position of the sharp jumpschanged between the measurement here (it was done first) and that presented inFig. 6.2. This change was probably the result of a charge reconfiguration in theclose vicinity of CNT 2.

In panel (b) we plot Isw(Vg) measured across a jump. We notice that dIsw/dVg

changes sign. In panel (c) we plot the power dependence of Shapiro steps, whenapplying a microwave frequency close to the mechanical resonance at the gatevoltage indicated by the dashed line in panel (b). Next we tune the power of themicrowave source to the position of the dashed line in panel (c), and measure themechanical resonance as a function of gate. This is shown in panel (d).

We observe a striking dip in resonance frequency when the gate voltage is tunedacross the jump. Such dips have been observed before in suspended CNT res-onators with an embedded quantum dot that was weakly coupled to the leads,and are understood in terms of a strong coupling between electron dynamicsand mechanics [1, 10, 11]. In our resonator (CNT 1) Coulomb blockade is al-most completely absent, the device is in the open regime where the conductanceis dominated by Fabry-Perot interference (see Sec. 2.1.3). In the open regimewe do not observe such dips, because charge quantization is absent there (seeSec. 5.3.2). In CNT 2 however charge quantization is not absent, and we canobserve its charge transitions because CNT 1 is capacitively coupled to CNT 2.Our data suggests that the transition from n to n+ 1 electrons in CNT 2 causesa resonance frequency dip in CNT 1, because CNT 2 capacitively induces fluctu-ating electron forces in CNT 1 that drive the resonator and cause the dip, as ifthe charge transition was in CNT 1 itself.

This is an atypical feature that is characteristic to our device, and is probablyhard to reproduce. Therefore we have done most of our experiments in a differentgate range, where CNT 2 has no effect on the conductance, or mechanics of CNT1. We do find this feature interesting and have studied it a bit more.

With the gate voltage tuned at the minimum of the dip (linecut 2 in panel (d))we find a signal with an amplitude of ΔV ∼ 2μV, but at the sides (linecut 1

157


-20

0

20

-21 -20 -19 -18-1.5

0

1.5 V( V)

Vg( V)

I(nA

)

-15 0 15Vg(mV)

Vg,0=-18.24V

Vg,

0

(a) (b)

0.6

0.85

1.1

I sw(n

A)

-4-2024

V( V)

-90 -60-0.5

0

0.5V( V)

P(dBm)

I(nA

)

-15 0 15Vg(mV)

(c) (d)

950

952.5

955

f(M

Hz)

-1

0

1

1 2 3

Figure 6.9: Characterization. (a) Measured V (I) as a function of Vg. The lightregion around I ∼ 0 is the supercurrent branch. (b) Zoom in on switching currentas a function of Vg. The dominant slope is the sharp jump indicated by the verticalline in panel (a). (c) Power dependence of Shapiro steps, f = 953.67MHz, taken atVg = −18.243V (vertical dashed line in panel (b)). (d) Mechanical resonance as afunction of Vg. Data taken at P = −64.16 dBm (indicated by dashed line in panel (c))and I ∼ 0 nA. Dashed lines indicate position of linecuts in Fig. 6.10.

and 3 in panel (d)) the signal amplitude is at ΔV ∼ 5 nV almost three orders ofmagnitude smaller, at the same RF power. These linecuts are shown in Fig. 6.10.We also note in this figure that the signal taken at linecut 3 has opposite signcompared to linecut 1, which suggests that it is due to mixing.

954.024 954.134 954.244

-4

-2

0

2

f(MHz)

V(n

V)

953.861 953.971 954.081-4

0

4

f(MHz)

V(n

V)

950 952.5 955

-1

0

1

f(MHz)

V(

V)

(a) (b) (c)

1 2 3

Figure 6.10: Linecuts taken near charge transition. (a) Signal as a function offrequency. Data is fitted to a Fano lineshape (grey line), fr = 954.134MHz, Q = 28.7k,and taken at Vg indicated by line 1 in Fig. 6.9d. (b) On the charge transition thelineshape is highly non-linear and the signal has a large amplitude (taken at line 2 inFig. 6.9d). (c) As in panel (a), but taken at line 3 in Fig. 6.9d, fr = 953.971MHz,Q = 29.6k. All data was taken at P = −64.16 dBm. We used a 25× longer integrationtime in panels (a) and (c) compared to panel (b).

158


Because the difference in slope of dG/dVg ∝ dIsw/dVg at the gate positionsindicated by linecuts 1-3 and linecut 2, is only one order of magnitude, it is notsufficient to account for the change in signal size. The lineshape taken at linecut2 looks highly non-linear and in combination with the large signal amplitudethis suggests that the resonator is driven in the non-linear regime by fluctuatingelectron forces (Coulomb forces) that are the result of capacitive coupling toelectron dynamics at the charge transition in CNT 2. This mechanism is similarto that presented in Ref. [10].

Fractional driving and detection at the charge transitionWe have measured the response of the mechanical resonator, while driving atinteger fractions of its fundamental mode, similar to Fig. 6.4, but now tuningthe gate voltage through the dip in mechanical resonance. Our data is presentedin Fig. 6.11. We observe a signal when driving at 1/2fr, 1/3fr, and 1/4fr. Atfractional driving the dip is also present.

950.0

952.5

955.0

f(M

Hz)

237.5

238.0

238.5

317.0

317.5

318.0

475

476

477

-15 0 15Vg(mV)

-15 0 15Vg(mV)

-15 0 15Vg(mV)

-15 0 15Vg(mV)

(a) (b) (d)(c)

f(M

Hz)

f(M

Hz)

f(M

Hz)

Figure 6.11: Fractional driving and detection at the charge transition. Weplot dV/df in all panels, and extract the mechanical resonance as a function of Vg drivingat integer fractions of the mechanical mode. The arrows indicate the resonance position.(a) Driving at 1/2fr. Data was taken at P = −60 dBm. (b) Driving at 1/3fr. Datawas taken at P = −30 dBm. (c) Driving at 1/4fr. Data was taken at P = −30 dBm.(d) Here we plot dV/df taken from the dataset presented in Fig. 6.9d. Superimposedare peak fits that were done on the mechanical resonance in panels (a,b,c). The y-axisof the overlays are scaled with respectively a factor 2,3 and 4. The current bias wasI ∼ 0.

In Fig. 6.12 we plot linecuts taken from the data presented in Fig. 6.11. We findthat the fractional resonances are at near-perfect integer fractions of fr.

159


953.007 953.507 954.007-20

-10

0

10

f(MHz)

V(n

V)

238.22 238.37 238.52-20

0

20

40

f(MHz)V

(nV

)

476.818 476.968 477.118-20

0

20

40

f(MHz)

V(n

V)

317.737 317.889 318.037-40

-20

0

20

f(MHz)

V(n

V)

(a) (b)

(c) (d)

Figure 6.12: Mechanical resonance at fractional driving frequencies. Weshow linecuts taken at the dashed lines (Vg = −18.247V) in the datasets in Fig. 6.11.Data has been fitted to a Fano lineshape. (a) P = −64.16 dBm, fr = 953.5072MHz,Q = 5.4k. (b) P = −60 dBm, fr,1/2 = 476.9680MHz, Q = 8.8k, fr/fr,1/2 = 1.9991.(c) P = −30 dBm, fr,1/3 = 317.8886MHz, Q = 7.9k, fr/fr,1/3 = 2.9995. (d) P =−30 dBm, fr,1/4 = 238.3699MHz, Q = 6.1k, fr/fr,1/4 = 4.0001.

6.4 Conclusion

We have investigated mechanical resonance when driving the system at integerfractions of a mechanical mode. By consideration of the applied power we findthat it is unlikely that excitation and detection is due to spurious higher har-monics of the RF source. We find a resonance signal at precise integer fractionsof the mechanical mode, which indicates detection of parametric driven motion.As a function of magnetic field we find a sign change of the resonance lineshapethat indicates that the signal is due to mixing of mechanics with AC Josephsoncurrents, as has been discussed in the previous chapter.

Due to the nature of our device Josephson harmonics are likely candidates to mixdown parametrical excited motion to a DC signal. The magnetic field dependencetaken at fractional driving shows that Josephson mixing is also involved there,which is consistent with our hypothesis of mixing with Josephson harmonics.

Our experiments in the direct vicinity of the sharp conductance jump indicatethat the motion of the superconducting nanotube is strongly excited by Coulombforces that are due to a strong coupling to electron dynamics in a second nan-otube that is close to the superconducting nanotube. In this regime we have alsodetected a signal at fractional driving.

160

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[2] A. Eichler, J. Chaste, J. Moser, and A. Bachtold, Parametric amplificationand self-oscillation in a nanotube mechanical resonator, Nano letters (2011).



[5] D. Garcıa-Sanchez, A. San Paulo, M. Esplandiu, F. Perez-Murano, L. Forro,A. Aguasca, and A. Bachtold, Mechanical detection of carbon nanotuberesonator vibrations, Physical review letters 99(8), 85501 (2007).

[6] Rohde&Schwarz, R&S SMB100A RF and Microwave Signal Generator Spec-ifications, 2012.

[7] V. Deshpande and M. Bockrath, The one-dimensional Wigner crystal incarbon nanotubes, Nature Physics 4(4), 314–318 (2008).

[8] F. Kuemmeth, S. Ilani, D. Ralph, and P. McEuen, Coupling of spin andorbital motion of electrons in carbon nanotubes, Nature 452(7186), 448–452 (2008).

[9] M. Jol, Physics of very small bandgap carbon nanotube quantum dots, MSc.thesis, Delft University of Technology, 2010.



161

Chapter 7

Future directions forsuperconducting CNTresonators

7.1 Current status and main challenges

Clean CNT resonators have outstanding properties that are not found in othermechanical resonators. Their small mass, high stiffness and few defects allow forflexural modes with mechanical resonances at GHz frequencies while maintainingquality factors on the order of 105. The mode of a resonator with fr > 1GHzcan in principle be cooled to its quantum ground state in a dilution fridge atT < 50mK.

One challenge in this field is to make a detector that can probe the zero pointmotion fluctuations of the CNT resonator. Those fluctuations set the limit of thesensitivity to small forces. When the level spacing of a superconducting qubit ismade sensitive to the displacement of a mechanical resonator, qubit spectroscopycan be used to detect its displacement.

Another challenge is to study the quantum nature of the CNT resonator. Fun-damental questions in quantum mechanics can possibly be addressed in suchsystems. The “common” way to demonstrate that an object is quantum-enabled,is to bring it in a superposition state. Since superconducting qubits and mechan-ical resonators can operate in a similar frequency regime, and their technology isin principle compatible, it was feasible that a superposition state of a microwavephonon and a photon could be made, if they could be coupled strongly enough [1].Such device is a quantum-coherent photon-phonon interface, where the resonatorcan for example be used as a quantum memory of qubit states [2].

In 2010 this system was first realized by O’Connell et al. [3]. In their systemthe resonator was coupled to a phase qubit, and strong coupling was achievedthrough the effect of motion on the phase of the qubit. One limitation of theirsystem was the relatively short lifetime of the superposition state, limited by the

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mechanical quality factor: Q ≈ 260.

Very recently an aluminum drum resonator was coupled to a transmon qubitby Pirkkalainen et al. [4]. The resonance frequency of the mechanical resonatorwas 72MHz, and the quality factor was Q ≈ 5.5k. Their device was embeddedin a hybrid system consisting out of a LC resonator, a transmon qubit and amechanical resonator. They were able to reach strong coupling between thequbit and mechanical resonator, which allowed for the creation of photon-phononsuperposition states in the resolved sideband regime.

The aluminum drum resonators are now among the most successful systems toachieve strong coherent coupling between microwave photons and phonons. Theyhave typical quality factors of ∼ 3 · 105 [5], and in that sense they are similar toour CNT resonators. Their resonance frequency is however usually on the orderof 50MHz, which makes “direct” strong qubit-resonator coupling not possible.

As CNTs have Q-factors that are comparable to that of the aluminum drum res-onators, they in principle allow for low decoherence rates1, and allow for longermemory times compared to the device by O’Connell et al. The CNT resonatorfrequency can be in the same regime as a transmon qubit transition frequency.Also, our CNT resonators are fundamentally different from aluminum drum res-onators, because the weak-link is the mechanical resonator. We will show in thischapter how we can take advantage of this.

In our work we have for the first time developed the technology to make highquality clean CNT resonators compatible with superconducting devices. On thisnew platform clean CNT resonators can potentially be integrated with supercon-ducting LC resonators and superconducting qubits.

Our suspended CNT Josephson junctions allow for a superconductivity enhancedmixing detection of the CNT motion, that we attribute to the AC Josephsoneffect. This enhancement allows us to detect CNT motion that is on the order ofthe thermally excited motion at T ∼ 90mK, a sensitivity that (as far as I know)has not been reported before.

Quantum dots form naturally in carbon nanotube resonators, and as a conse-quence the conductance through such system is strongly depended on the chargeon the resonator. The capacitive coupling of the resonator to a gate in its vicin-ity enables detection of mechanical motion by mixing or rectification techniques.While this detection method relies only on charge (or capacitive) coupling, adifferent method that relies on flux (or inductive) coupling is also of importance.

Independently of our experiments, another pioneering experiment on a suspendedclean CNT resonator SQUID device has recently been reported by Schneideret al. [7]. A strong responsivity of flux to static displacement was found. Inthese devices superconductivity allows for the read-out of mechanical motion byinductive coupling rather than capacitive coupling.

In the following sections we will discuss some possible follow up experiments inwhich the motion of CNT resonators is coupled to superconducting devices. One

1Decoherence is the irreversible loss of a quantum superposition, and can be quantified bythe contrast of quantum Rabi oscillations [6].

7.2. Coupling CNT motion to a transmon qubit

immediate obstacle to build such devices is the incompatibility of the low-yieldfabrication process reported in this thesis, with the proposed experiments. Thenew stamping technology [8] that we discussed in Sec. 1.5, could be used todirectly stamp good (low bandgap) CNTs in place, on a chip with predefinedsuperconducting devices.

We will discuss CNT motion coupled to a transmon qubit and to a superconduct-ing LC resonator. Then we will discuss a Josephson parametric amplifier with anembedded superconducting CNT resonator. We will conclude by suggesting ex-periments in which high-field compatible CNT Josephson junctions can be used,including an experiment on P1-centers in diamond.

7.2 Coupling CNT motion to a transmon qubit

In this section we will first give a brief overview on superconducting qubits.This discussion is based on the introduction chapter of Ref. [9]. A review onsuperconducting qubits can be found in Ref. [10]. Interaction between NEMS andsuperconducting qubits is a growing topic of (mostly theoretical) investigation, anearly proposal and recent review can be found in Refs. [11,12]. In the second partwe will discuss how a CNT resonator changes the basic properties of a transmonqubit, and we will estimate the coupling strength.

Superconducting qubitsIn superconducting quantum circuits charge q and flux Φ (or phase ϕ), are conju-gate variables. The family of superconducting qubits can be divided into chargeand flux/phase qubits, depending on the ratio of the charging energy and theJosephson energy: EJ/Ec. Examples of the charge qubit (EJ/Ec � 1) arethe Cooper pair box and the quantronium. The phase qubit, flux qubit andtransmon are examples of qubits where the junctions are in the phase regime,EJ/Ec � 1. Sensitivity to charge noise was a major source of decoherencein charge qubits. In a transmon qubit this charge sensitivity is largely re-duced [13, 14], which has resulted in record coherence times compared to othersuperconducting qubits [15, 16].

We have chosen to discuss the transmon qubit here, because it combines longcoherence times with typical Josephson energies that can (in principle) be pro-vided by a CNT Josephson element. In the experiment by Schneider et al., arelatively big switching current of Isw = 10nA was found using contacts from aReMo alloy with Tc ∼ 5.5K [7]. The Josephson energy of these junctions is (atleast) EJ ∼ 20μeV, which is only a factor of four below that of the Josephsonelements typically used in transmon qubits.

Interaction between a CNT resonator and a transmon qubitThe quantum nature of a (quantum) mechanical resonator can be studied by cou-pling it to another quantum system, for example to a superconducting qubit [1,3].We distinguish two types of coupling: σx,y coupling, which allows for the hy-bridization of energy levels and superposition states, and σz coupling, whichshifts the energy of the qubit states. The term “coupling” is mostly used for σx,y

coupling. We will follow the usual notation and keep g for σx,y coupling, and we

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7. Future directions for superconducting CNT resonators

introduce the parameter gz for σz coupling.

When the coupling g is sufficiently strong, coupled quantum systems can bemodeled by the Jaynes-Cummings Hamiltonian [17]. This model describes thecoherent interaction between a harmonic oscillator and a two-level system. Orig-inally it was derived for atoms in an optical cavity, but it can also be applied tosuperconducting qubits coupled to SC waveguide resonators and nanomechanicalresonators [3, 18].

The σx,y interaction strength is described by the coupling energy �g. Here gcorresponds to the rate at which the energy of one system can flow to the otheron resonance. Depending on the size of g in comparison to the relaxation rate ofthe qubit (γ = 1/T1, where T1 is the relaxation time), and the resonator dampingrate (κ = fr/Q, where fr is the resonator frequency, and Q is the quality factor),the system is either in the weak or strong coupling regime [9].

Coherent interaction takes place in the strong coupling regime, where the rateof energy exchange g is faster than the loss rates of the harmonic oscillator andthe qubit: g > κ, γ. In this regime a photon emitted from the qubit into theresonator can be recovered after an interaction time Δt = π/Ω, and quantumcoherence is preserved within the coherence time 1/κ and 1/γ. Here Ω = 2g/hcorresponds to the Rabi-swap frequency at which the exchange takes place [3].

For quantum-coherent experiments the main challenge is to design an experi-ment where the motion of a nanomechanical resonator is strongly (σx,y) coupledto a qubit. This has been successfully implemented, with a phase-qubit thatwas strongly capacitively coupled to a piezoelectric dilatation resonator [1, 3].The nature of the system was such that the motion of the resonator altered thecurrent bias (phase) of the qubit. This resulted in a strong σx,y coupling, andsuperposition states could be made in the resonant regime where �ωr = E01.

By introducing a CNT resonator in a transmon qubit architecture, it is not yetclear how a strong σx,y coupling can be achieved. The resonant regime can inprinciple be achieved with a ∼ 2.8GHz CNT resonator, and a transmon qubitsystem in the same frequency regime (see Fig. 7.1). Because strong σx,y couplingis in principle not present, no hybridization is expected when the two systemsare on resonance.

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2.832 2.833 2.834

106

114

I(pA

)

T<<hf/kB (130mK)

Q=60kT=50mKzpm=0.5pm

frequency (GHz)0.350.30Flux, Φ/Φ0

0.42

ω+I

ω-I

ωQ

ω-III

ω+III

2.95

2.90

2.85

2.62

2.57

erFeuqn

,ycω

2/π

G(Hz)

ms=0

ms= ±1+1

-1

ω-

ω+2.88GHz

IIII

B N-V

F

c

(a) (b)

Figure 7.1: Compatible systems: RF waveguides, superconducting CNTs,diamond NV-centers, transmon qubits. (a) Mechanical resonance of a fr ∼2.8GHz CNT resonator suspended on superconductors (our data). Measured in oneof the first generation devices, discussed in Sec. 5.4.1. This mode is the fundamentalflexural mode, or one of the first harmonics (data not shown). (b) Qubit spectroscopyon a hybrid system with diamond NV-centers, transmon qubit and RF waveguides all onresonance at f0 ∼ 2.8GHz. Anticrossings indicate strong coupling between NV-centerstates and qubit states. Panel (b) adapted from Kubo et al. [19].

We foresee however a significant σz-type coupling. This could be detected in aspectroscopy experiment, where static CNT displacement is to be observed as achanging qubit gap. Fluctuations of the CNT position at its resonance frequencycan in principle modulate the qubit gap, which could be measured in a Ramseyfringe experiment on the qubit [20]. We suggest two measurements:

� Measure the free evolution coherence time T2 from Ramsey fringes mea-sured while driving the CNT resonator with an additional microwave tone.Increasing the amplitude of the CNT resonator fluctuations, the “noise” itadds to the qubit gap will increase the dephasing of the qubit, lowering T2.

� Measure the linewidth of the qubit resonance as function of the CNT res-onator driving power, increasing noise can result in a broadening of theresonance peak. For these experiments the CNT resonator does not haveto be on resonance with the qubit. One complication would be to distin-guish noise added to the qubit by the resonator, from noise added by themicrowave tone used to drive the resonator.

In the remainder of this section we will discuss two ways to obtain σz couplingin a transmon-CNT system. First we will derive a convenient expression for thesensitivity of the qubit level spacing to the Josephson energy.

Sensitivity of transmon qubit level spacing to Josephson energyThe level spacing of a transmon qubit is given by [13]:

E01 ≈√

8EJEc , (7.1)

where EJ is the Josephson energy of the Josephson element, and Ec is the charg-ing energy that is mainly determined by a designed on-chip capacitance. By

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embedding a pair of junctions in parallel (a SQUID) in a transmon qubit design,its Josephson energy can be made sensitive to flux [14, 21–23]. This allows for a∼ 1GHz tunable energy splitting.

In Fig. 7.2a we present a SEM picture of a state-of-the-art transmon qubit fab-ricated at NIST in Boulder, and in panel (b) we present a sketch of a futuristictransmon device with a suspended CNT Josephson junction.

Vg

(a) (b)

/2 resonator: for qubit readout

capacitive shunt JJ1 JJ2

Figure 7.2: Suspended CNT transmon qubit. (a) SEM picture of a transmonqubit, adapted from Ref. [23]. (b) Schematic of a futuristic transmon qubit with onesuspended CNT Josephson junction (at right). The tension and charge on the CNT canbe tuned by a voltage Vg on a gate in the vicinity of the CNT.

A small change δEJ shifts the transmon qubit energy by:

δE01 =E01

2

δEJ

EJ. (7.2)

Displacement of a suspended CNT Josephson junction can result in a small changeof its Josephson energy mediated by a change of flux, or by a change of charge.The size of this gz coupling can be compared to the relaxation rates.

The relaxation rate of a 2GHz resonator with a Q = 105 is κ = fr/Q = 20 kHz.The relaxation rate of state-of-the-art transmon qubits with an energy splittingof 7GHz, is γ = 1/T1 ≈ 1/50μs ≈ 20 kHz as well [15,16]. A coupling gz > 20 kHzwill bring the system in the strong gz coupling regime.

We will now estimate the gz coupling from the change of the Josephson energyto a displacement induced change of flux and charge.

Motion-flux coupling in a transmon qubitSchneider et al. experimentally find that the responsivity of flux to displacementin a suspended SQUID resonator is large enough to reach the strong gz couplingregime [7]. Typical parameters of their device are: fr = 126MHz, Q = 3 · 104and Isw = 10nA. The flux responsivity is 0.35mΦ0/pm at B = 1T. Assuming atransmon qubit with a gap of E01 = 6GHz, they obtain a zero-phonon couplinggz = δEzpm

01 = 7MHz, gz � κ, γ. This indicates that the strong gz coupling limitcan in principle be obtained by coupling to flux, but this also implies that thetransmon qubit has to be operated at 1T.

Motion-charge coupling in a transmon qubitIn suspended CNT resonators the Josephson energy also depends strongly on theconductance (charge) on the CNT (see Sec. 2.5.2). We will now estimate the

168


responsivity of charge to displacement in our devices, and use this to estimatethe gz coupling to a transmon qubit.

As Cg ∝ y (see Eq. (B.4)) we can estimate the capacitance change due to thezero point motion as follows:

yzpm

y=

δCg

Cg=

1.5 pm

300 nm= 5× 10−6 , (7.3)

where y is the distance of the CNT to the gate, and yzpm has been estimatedusing Eq. (5.2), and the typical parameters of our device.

The zero point motion fluctuations result in charge fluctuations of order:

δqzpm = δCgVg = 5× 10−6 × 0.5 aF× 10V = 1.6× 10−4e , (7.4)

where we used Cg = 0.5 aF (see Tab. A.1), and Vg = 10V.

The responsivity of Josephson energy to charge can be estimated from Isw(Vg),as is shown in Fig. 5.5. We find:

∂EJ

∂q=

0.3 nA�/2e

2e= 0.3μV , (7.5)

which results in δEzpmJ = 0.3μV × 1.6 × 10−4e = 5 × 10−5 μeV. This we use to

estimate:

gz = δEzpm01 =

E01

2

5× 10−5 μeV

2μeV≈ 75 kHz , (7.6)

where we have taken E01 = 6GHz.

This charge coupling is two orders of magnitude smaller then the flux couplingfound by Schneider et al. We expect the charge coupling in their device also to belarger, because the gate capacitance is an order of magnitude larger, Cg ≈ 5 aF.Also the zero point motion is about a factor two larger in their device. We willnow calculate the coupling strength to charge, for the device studied by Schneideret al.

The large gate capacitance results in a responsivity of Josephson energy to chargethat is about one order of magnitude larger, ∂EJ/∂q = 6nA/2e(�/2e) = 6μV,and charge fluctuations that are about one order of magnitude larger as well,(3.6 pm/300 nm)×5 aF×10V = 3.7×10−3e. With these numbers we get δEzpm

J =6μV × 3.7× 10−3e = 2.2× 10−2 μeV, which is about three orders of magnitudelarger compared to our device, and results in a coupling gz ≈ 7MHz (takingEJ = 20μeV). This indicates that the strong gz coupling regime can be achievedby coupling to charge as well as flux.

The advantage of using a longer CNT resonator is that the coupling is strongerand the zero point motion is larger. The obvious disadvantage is that the fun-damental bending mode is at a low frequency, which makes additional cooling(below dilution fridge temperatures) necessary to achieve the quantum regime.

Our devices have fundamental modes that are with 1GHz about one order ofmagnitude higher. Our estimations indicate that the strong gz coupling regime

169


can be just reached in our devices, but it sets high demands on the dampingrates of qubit and resonator. An important advantage of coupling to chargerather than flux, is that operation at B = 0 is possible. A gate voltage can beused to increase the coupling by about one order of magnitude, reaching gz ≈0.5MHz. Summarizing, to reach the strong gz coupling regime by responsivity ofJosephson energy to charge one could use CNT resonators as Josephson elementin a transmon qubit architecture, and make the CNT device compatible withlarge gate voltages.

7.3 Coupling CNT motion to superconductingLC resonators

Sideband coolingInstead of cooling mechanical oscillators in a dilution fridge, a different approachthat involves active cooling has been implemented recently to reach the groundstate of a 10MHz resonator in a dilution fridge at 25mK [2], and a 3.7GHzresonator in a flow cryostat at 20K [24]. Quantum-coherent coupling and near-ground state cooling of a 78MHz resonator was achieved in a 3He cryostat at650mK [25].

These resonators were actively cooled by coupling the resonator to a microwave [2],or optical cavity [24,25]. This cooling method results from the coupling betweenphotons and phonons (σx,y-like coupling), and is called sideband cooling [26–30].In this section we will estimate the coupling strength g, between a CNT resonatorand a superconducting LC resonator. First we will discuss sideband cooling, andgive examples of a nanowire resonator and a drum resonator embedded in amicrowave cavity [5, 31].

An electro-mechanical microwave cavity is equivalent to an optical cavity with amovable mirror. In such cavity the resonance frequency is determined by the dis-tance between the mirrors, and altered when one of the mirrors moves. Photonsbounce back and forth between the mirrors and exert a radiation force Frad onthem (see Fig. 7.3a). The average force on a mirror is proportional to the numberof photons in the cavity, nc. This number depends on the driving frequency ωd,and the cavity resonance frequency ωc, which is very sensitive to a displacementof the mirrors (see Fig. 7.3b). As a result, Frad depends on the position of themirrors. The radiation force does not respond instantaneously on displacement,because nc can only change at a rate ∼ κ. As we will see, this time-lag is crucialto cooling by radiation force.

The radiation force as a function of mirror position has a Lorentzian shape (seeFig. 7.3c) [30]. To cool the mirror it should be positioned on the left slope (thisis done in practice by red detuning the drive laser). The mirror position willmove back and forth, due to thermal fluctuations or driven motion. Since theradiation force lags behind on the position, it will experience a smaller force thanexpected when approaching the resonance, and a larger force while retracting.This is indicated by the blue ellipsoid in panel (c). On average the radiation

170

7.3. Coupling CNT motion to superconducting LC resonators

input laseropticalcavity

mirrormm

reflectedpower

(a) (b) (c)

Figure 7.3: Sideband cooling in a resonator cavity system. (a) Schematic of anoptomechanical setup. (b) Intensity of light in the cavity. The cavity mode can be tunedby the resonator position u. At a red detuned excitation the resonator displacement canenhance the intensity in the resonator. The intensity is proportional to the number ofphotons in the cavity. (c) The radiation force Frad exerted on the resonator lags behindon u by the cavity hold time 1/κ. This results in an effective detuning-depended coolingand heating cycle, indicated by the ellipsoids. Figures adapted from Refs. [30, 32].

force extracts work from the mirror:

∮Fraddu < 0. This is an extra damping

term that effectively cools the mirror. The opposite effect is reached when thedriving is blue detuned (indicated by the red ellipsoid) [30].

In the quantum mechanical picture there is an optimal detuning drive frequencyat ωd = ωc − ωm. At this frequency a photon will be preferentially up-convertedto ωc, because the photon density of states (cavity intensity) is maximum there.This process can be facilitated by the extraction of a phonon from the mechanicalresonator, and is efficient in the resolved sideband regime, where κ < ωm (thebest displacement sensitivity is achieved when κ � ωm). The motion of theresonator is damped, and it can be detected by monitoring photons leaving thecavity [2, 33].

The signal due to the zero point motion is typically very small and has to beamplified with an amplifier that adds no noise. Josephson parametric amplifiersare suited very well for this purpose and will be discussed in Sec. 7.4. In typicalsetups a HEMT (high electron mobility transistor) at 4K is used to amplify theoutput of a JPA (Josephson parametric amplifier).

Interaction Hamiltonian and σx,y-like couplingOptical cooling of mechanical nanomechanical resonators can be efficient in theresolved sideband regime. In this regime the interaction Hamiltonian (similar tothe Jaynes-Cummings Hamiltonian) is given by [2, 25, 34,35]:

H = �g(ab† + a†b) . (7.7)

Here g is the field enhanced coupling rate: g =√ng0, where g0 is the vacuum

coupling rate, and n is the average photon number in the cavity. This couplingis off-diagonal (σx,y-like), and allows to swap a phonon with a photon at a rateΩc = 2g. In case Ωc > κ, γ the coupling is quantum-coherent. The couplingstrength can be tuned with the intensity of the laser or microwave drive.

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Capacitive coupling a mechanical resonator to an LC cavity modeWhile we have taken an optical cavity as an example for sideband cooling, ananalogue discussion holds for a microwave cavity such as a superconducting LCresonator. In Fig. 7.4 we show two implementations of mechanical resonatorsembedded in an LC circuit.

(a) (b) (c)

2 m

Figure 7.4: LC resonator coupled to mechanical resonator. (a) Schematic of aLC resonator coupled to the motion of a mechanical resonator with spring constant km.(b) Aluminium nanobeam resonator embedded in a LC resonator, properties of beam:fr = 237 kHz and Q = 2.3k, properties of LC resonator: f0 = 5GHz and Q = 10k. (c)Aluminum nanodrum resonator embedded in a LC resonator, properties of the drum:fr = 10.7MHz and Q = 360k . Figures adapted from Refs. [5, 31,33].

Here the mechanical resonator is capacitively coupled to the LC circuit. Thisis shown schematically in panel (a), a displacement of the resonator results in achange of the LC capacitance, which changes the resonance frequency. In panel(b) the mechanical resonator is a nanowire resonator, and in panel (c) it is a drumresonator. The single-photon coupling strength is determined by the change ofcavity frequency due to the zero point motion of the mechanical resonator. Thechange of cavity frequency due to a capacity change by a displacement y is givenby [5]:

G =∂ω0

∂y=

ω0

2Ct

∂Cg

∂y, (7.8)

from which the single-photon coupling strength g0 = Gyzpm, and the single-photon coupling rate g0/2π, are determined. For a given yzpm, strong couplingis achieved by maximizing G, which is eventually determined by the capacitanceparticipation ratio, i.e. the relative change of capacitance of the LC circuit dueto the zero point motion of the resonator. In Tab. 7.1 we have summarized someparameters, and we compare nanowire and nano-drum resonators with a CNTresonator.

It turns out that strong coupling (g > κ, γ) is easier achieved in drum-resonatorscompared to nanowire-resonators, because the capacitance participation ratio isup to three orders of magnitude larger in this system. Due to their small sizeand low mass, CNT resonators have large zero point motion fluctuations, but thechange of capacitance caused by them is so small that only very small couplingcan be achieved. The strong coupling regime has been achieved with the Aldrum resonators [4, 5, 37, 38]. By driving those systems, g was increased fromresp. 410Hz to 1MHz, 44Hz to ∼ 1MHz, and 20Hz to ∼ 5MHz exceedingthe decoherence rates of cavity and resonator. In this regime the resonator by

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7.3. Coupling CNT motion to superconducting LC resonators

Table 7.1: Coupling of mechanical resonators to superconducting LC resonators,w=wire, d=drum.

ω0/2π Ct ∂Cg/∂y G/2π yzpm g0/2π δCzpm

(GHz) (fF) (aF/μm) (kHz/nm) (fm) (Hz) (zF)

Al w [31] 4.9 359 170 1.16 40 0.05 7 · 10−3

Al w [36] 7.5 - - 32 27 0.9 -SiN w [34] 7.5 260 6 · 103 84 26 2.2 2 · 10−1

Al d [5] 7.5 380 1 · 106 1 · 105 4.1 410 4Al d [37,38] 7.0 24 13 · 103 2 · 103 22 44 0.3Al d [4] 4.8 61 26 · 103 1 · 103 20 20 0.5

CNT, cap. 7.5 300 2 3 · 10−2 1500 0.04 3 · 10−3

CNT, ind. 7.5 - - - 1500 6000 -

Teufel et al. was cooled to its ground state [2, 5], the resonator by Massel et al.was cooled close to its ground state [38] and the resonator by Pirkkalainen etal. was coherently coupled to a transmon qubit [4]. To reach strong coupling inthese systems the driving power was such that there were roughly ∼ 108 coherentphotons in the LC resonator.

Inductive coupling a CNT resonator to an LC cavityOur CNT resonators are unique in the sense that the size of the supercurrent isdetermined by the charge on the CNT, which is a function of the CNT position.At small phase bias ϕ ≈ 0, the effective Josephson inductance of a Josephsonjunction is determined by the supercurrent [10]:

LJ0 = (Φ0/2π)/Ic . (7.9)

If we take Ic = Isw ≈ 1 nA, in our junctions: LJ0 ≈ 300 nH. Noting that the“real” critical current can be an order of magnitude larger [39], the inductance canbe an order of magnitude smaller as well. We also note that switching currentsof ∼ 10 nA were achieved by Schneider et al., which indicates that a Josephsoninductance of ∼ 10 nH is feasible for CNT junctions.

Typical LC circuits with ω0/2π ≈ 7.5GHz have L ≈ 10 nH and C ≈ 300 fF (seeTab. 7.1). This inductance can in principle be provided by a CNT junction. Sincethe inductance is determined by the supercurrent, which changes when the CNTmoves, the LC resonance depends on the CNT position. Using Eq. (7.5), we find:

δIzpmsw =0.3 nA

2e× 1.6× 10−4e ≈ 2.4× 10−5 nA . (7.10)

For our junction with Isw = 1nA, δLzpmJ0 ≈ (Φ0/2π)/1 nA − (Φ0/2π)/(1 +

10−5) nA ≈ 3 pH. For an LC resonator with C = 300 fF, this implies a change ofresonance frequency of:

δωzpm0 /2π = (LJ0C2π)−

12 − ((LJ0 + Lzpm

J0 )C2π)−12 ≈ 6 kHz . (7.11)

This coupling is orders of magnitude larger than the capacitive coupling. Thephysical reason for this is that the inductance participation ratio: 3 pH/10 nH =

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3 · 10−4 is roughly six orders of magnitude larger than the capacitance partici-pation ratio: 3 · 10−3 zF/300 fF = 1 · 10−11 (see Tab. 7.1). It is perhaps possibleto bring this system in the strong coupling regime, by driving it with an intensemicrowave signal similar to the experiment on the disk resonator [2]. If this ispossible then also an experiment similar to that of Ref. [4] can be pursued. InFig. 7.5 we show an example of a superconducting resonator, and indicate wherea CNT Josephson junction could be added.

L L

150 m

r=( L C)-1/2

(a) (b)

LJ0+LJ0,y C

C

Figure 7.5: LC resonator with embedded suspended CNT Josephson junc-tion. (a) Schematic diagram of the LC resonator. The suspended CNT Joseph-son junction is modeled as an inductance that depends on the CNT position abovea gate. (b) Example superconducting LC resonator fabricated by Forn-Diaz et al.,with (total) C ≈ 250 fF, and (total) L ≈ 1.5 nH, resulting in a resonance frequencyof ωr/2π ≈ 8.2GHz [9]. A CNT Josephson junction could be embedded in such LCresonator, contributing to the total inductance.

LC resonator used to probe the current-phase relation of a CNT Joseph-son junctionWe will now suggest an interesting experiment that is unrelated to nanomechan-ics. The setup discussed in Fig. 7.5 also allows one to map the current-phaserelation of the suspended CNT Josephson junction. Such experiment has beendiscussed in Ref. [40]. The phase dependence of the Josephson inductance canbe inferred by measuring the resonance frequency of the LC circuit as a func-tion of phase-bias. Knowledge of the current-phase relation is of fundamentalinterest, to help understand the microscopic origin of supercurrent in the CNT.Such measurements can perhaps also detect the “real” critical current, which isusually thought of to be about one order of magnitude larger than the switchingcurrent [39].

7.4 Josephson parametric amplifier with a sus-pended CNT junction

In parametric amplifiers the non-linear, time-varying reactance of a resonator isused to amplify an input signal. In case a reactive parameter oscillates at twice

174

the resonance frequency, energy can be pumped in or out of an input mode, eitheramplifying or damping it. This is similar to the child on a swing, amplifying ordamping its amplitude by shifting its center off mass at twice the swing frequency.An intense driving signal (pump) at ω can provide parametric oscillations at2ω, in case the nonlinear reactance is proportional to the intensity rather thanthe amplitude of the pump signal. This is the case for the current-dependentinductance of a Josephson junction [41,42].

Josephson parametric amplifiers (JPAs) are made with Josephson junctions andare operated in a dilution fridge. This makes them especially suited to amplifymicrowave signals originating from on-chip mechanical resonators. They are alsoeasily integrated with microwave cavities. With the detection of the zero pointmotion of a CNT resonator in mind, it is interesting to consider a JPA with anembedded CNT resonator as Josephson element. In this section we will presenta brief discussion of JPAs and share our basic ideas on how a CNT could bedirectly embedded in a JPA.

An extensive discussion on a new class of tunable JPAs can be found in the thesisof M.A. Castellanos-Beltran [42].

Displacement detection below the standard quantum limitJPAs are linear amplifiers that can (in principle) amplify the amplitude of amicrowave signal without adding any noise. A typical input signal is a sinusoid. Insuch a signal, information is encoded in amplitude or phase (the two quadraturesof the signal). These are conjugate variables. While quantum mechanics limitsthe combined precision with which two conjugate variables can be measured,it does not limit the precision with which one of the two quadratures can bemeasured [42].

With a phase-sensitive amplifier like the JPA, it is (in principle) possible toamplify one quadrature, while de-amplifying the other quadrature, and addingless than half a noise quantum [43]. Using JPAs the displacement of a mechanicalresonator can be detected with an imprecision that is smaller than the standardquantum limit (SQL) scale 1. With such detector it is possible to resolve thezero-point motion of the mechanical resonator.

Tunable Josephson parametric amplifiersTypical JPAs have two main disadvantages: They combine a small dynamicrange with a narrow bandwidth gain around a center frequency. Recently tunableJPAs have been built which can be operated with their center frequency tunablein the 4 . . . 8GHz range [41, 44, 45]. The center frequency can be shifted bytuning the flux through one or more SQUIDs that form the amplifier. Suchamplifiers have been successfully used to amplify the zero point motion signalfrom a nanomechanical resonator [2]. We will first briefly discuss the tunableparametric amplifier and then show two implementations.

In Fig. 7.6a we give a schematic illustration of a basic measurement system witha JPA. The Josephson elements form the inductive part of an LC resonator with

1This scale is set by the Heisenberg uncertainty principle, which ensures that an increasinglyprecise measurement of a harmonic oscillator is accompanied by an increasingly large backactionforce acting on the resonator. In the SQL the backaction force and imprecision both add �ωm/4of noise energy [36].


a resonance frequency at ω0, and a linewidth γ. The amplifier is driven by apump signal at ωp, tuned closely to ω0: ωp � ω0+δω. Here δω ∼ γ. This type ofoperation is called the doubly-degenerate operation mode, or four-wave mixing.This configuration is discussed in Refs. [41, 45].

( p- 0)/ 0

refle

cted

sig

nal

phas

e

-0.02 -0.01 0 0.010

2

Jose

phso

nph

ase

/2

p/ 0

0.95 1 1.05

(c)

0

0.05

0.1

Irf

Irf

signal in

JBA

Vp(t)

Zc

Zc

Rs=Zc

output

(a)

JPA

pump,Irfsin( pt)

0Irfsin( pt+ (I0,Irf))

I0

p is

s

Pou

tput

(b)

( s- p)/-2 -1 0 1 2

Gai

n (d

B)

0

20

40

Irf

Figure 7.6: Tunable Josephson parametric amplifier. (a) Schematic of a mea-surement system with a JPA (inside the dashed box). (b) The gain vs. detuning andpump amplitude. The center frequency ω0 can be tuned by flux threading the SQUIDloop. (c) Top panel: Josephson phase as function of pump frequency and amplitude.Bottom panel: Reflected signal phase. Figures adapted from Ref. [42].

The reflected signal contains the amplified input signal at ωs and an additionalimage signal at ωi = 2ωp − ωs. The input signal can be superimposed on thepump signal, or transduced by a modulation of the Josephson current (resultingin a modulation of ω0).

At the heart of the parametric amplification lies the non-linearity of the Josephsonelements. The equation of motion of the Josephson phase can be described bya Duffing equation (like Eq. (2.41)), which results in the lineshapes shown inthe top panel of Fig. 7.6c. Operating in the non-linear regime, the reflectedsignal phase is very sensitive to small changes of the resonator frequency ω0

(bottom panel). The pump frequency is tuned such that the device is operatedin the region indicated by the ellipsoid. At higher driving power the mechanicalresonance is bistable and the amplifier can be used as a bifurcation amplifier. Suchamplifier is a latching, or threshold amplifier, and is suitable for measurement onsuperconducting qubits [46], but less useful for amplification of displacement. Inpanel (b) we show the gain of a JPA as function of the detuning, ωs − ωp.

The amplifiers developed by Castellanos-Beltran et al. have a gain of 16 dBat a bandwidth of 2MHz. In a 1Hz bandwidth they can be used to amplifysignals in the −200 . . . − 130 dBm range by up to 25 dB (this corresponds to0.77�ω . . . 1.9 × 107�ω × 1Hz at ω = 2π × 8GHz). This is the dynamic rangeof the amplifier, which is determined by the critical power of the amplifier [44].The critical current determines the critical power above which the JPA goes inthe bifurcation mode, and the amplifier becomes a bifurcation amplifier.

JPAs with an embedded suspended CNT Josephson junctionIn Fig. 7.7a we show our proposal to embed a CNT resonator Josephson junctioninside a SQUID LC resonator. Displacement of the CNT can be transduced to a

176

7.4. Josephson parametric amplifier with a suspended CNT junction

change in the flux Φ threading the SQUID loop, or the Josephson current passingthrough the junction. The mechanical resonance frequency can be tuned by agate voltage Vg. This setup is similar to that discussed in Sec. 7.3, but herethe addition of an extra Josephson junction allows tunability of the resonancefrequency of the LC circuit. We propose to build the setup shown in Fig. 7.6a,and replacing the JPA in the dashed box by the schematic shown here in panel(a). The input signal is then replaced by the nanomechanical signal.

15 m

5mm

Vg

(a) (c)(b)

pump,signal output

signal input signal input

pump,signal output

Figure 7.7: JPAs with an embedded suspended CNT Josephson junction.(a) Schematic picture of a suspended CNT Josephson junction embedded in a JPA. (b)Dispersive magnetometer JPA by Hatridge et al. [45]. (c) JPA developed by Castellanoset al., the suspended CNT sketch is added by us [43].

In panel (b) we show a SQUID JPA that was developed by Hatridge et al. [45,46].This is exactly the implementation of the schematic drawn in Fig. 7.6a. TheSQUID is the heart of the amplifier. A signal on the flux line is transduced to achange of flux threading the SQUID, which modifies the Josephson energy andthe LC resonance. We propose to replace one of the SQUID junctions with asuspended CNT Josephson junction, and try to use it to amplify the CNT motion.

The best flux sensitivity Hatridge et al. report is 0.14μΦ0/√Hz at a 0.6MHz

bandwidth. With a bandwidth of 20MHz the sensitivity is 0.29μΦ0/√Hz [45].

Schneider et al. have estimated the flux due to the zero point motion in theirdevice to be 16μΦ0/

√Hz, which is about two orders of magnitude larger. This

implies that the Hatridge-type JPA is in principle suitable to detect the zeropoint motion fluctuations of such CNT.

One immediate complication of embedding a CNT junction is the relatively smallcritical current of the junction, resulting in a very small critical power as well.This results in a low saturation power, which prevents the amplifier for beinguseful (i.e. it can only amplify in the ∼ 0.5 . . . 10�ω range). The dynamic rangeof a SQUID JPA can be extended by increasing the number of SQUIDs. Thisapproach is shown in panel (c), where we show the SQUID-array JPA that wasdeveloped by Castellanos-Beltran et al. We have already discussed the excellentachievements of this amplifier. We propose to build such amplifiers, and add asuspended CNT junction in one of the SQUIDs and try to use this amplifier likethe Hatridge amplifier.

177


7.5 High magnetic field compatible CNT Joseph-son junctions

Superconducting devices and high magnetic fields are normally incompatible,because magnetic field breaks superconductivity (Sec. 2.6). Thin superconductingfilms, and type-II superconductors, can carry supercurrent up to fields of a coupleof Tesla (see Figs. 4.4 and D.1, and Ref. [47]). Josephson junctions that withstandsuch large fields are not trivial to fabricate. In this thesis we have reported twotypes of proximity-effect Josephson junctions that can carry supercurrent up toin-plane fields on the order of 1T (see Figs. 4.8 and 6.8). This property is dueto the choice of a low-dimensional weak link, and a contact with large criticalfield. In a field of B‖ ∼ 1T, the low-dimension nature of graphene and carbonnanotubes prevents Cooper pair-breaking by orbital depairing, and at these fieldseffect of Zeeman energy can become important (see Sec. 2.6.3). This has beenone important topic in our research, but the high-magnetic field compatibility ofCNT and graphene Josephson junctions has more applications. In this sectionwe will give a few examples in which we suggest that this can be used. We willfocus on CNT Josephson junctions.

CNT SQUIDs operating in high magnetic fieldsSQUIDs are very sensitive to magnetic fields, and can be used to detect mag-netization on a local scale, in for example a scanning SQUID setup [48]. Theoperation of such SQUIDs is limited to perpendicular fields B⊥ � 1T, but state-of-the-art SQUIDs can operate at in-plane fields B‖ � 7T [49]. Such deviceshave applications in the study of the properties of magnetic molecules.

SQUIDs that can operate at high perpendicular fields are of interest, for ex-ample to probe the spin-polarization of the 5/2 fractional quantum Hall state(FQHS) [50]. This state exists in the B⊥ = 2 . . . 10T range, and the measurementof its spin-polarization can shine light on the (non-Abelian) nature of this state.Recently a new diamond SQUID has been developed [51], that can be operatedat B = 4T, independent of the field direction. Such SQUIDs are likely candi-dates to be used to study the FQHS. Although the diamond films are fabricatedat 700�C, they can possibly be integrated on a chip with other superconductingdevices.

We suggest that CNTs can be used to make SQUIDs that are compatible withB⊥ ∼ 1T, using a type II superconductor as contact material. Furthermore, CNTbased SQUIDs offer certain advantages over diamond SQUIDs, since they can inprinciple be stamped in place on a pre-fabricated wafer, and are gate-tunable.

CNT JPAs operating in high magnetic fieldsJosephson parametric amplifiers in which all non-linear Josephson elements aresuspended CNT Josephson junctions, can possibly also be operated at large mag-netic fields. This would allow for the detection of small microwave signals at highfields, which can be of use in the detection of microwave radiation emitted byJosephson junction with Majorana states.

Signatures of Majorana states have been reported by our group in hybrid supercon-ductor-nanowire devices [52], at fields of B ∼ 250mT. An important future chal-

178

7.5. High magnetic field compatible CNT Josephson junctions

lenge is to probe the sin(4π) periodicity [53] of the current-phase relation of theMajorana state. DC voltage biased Josephson junctions emit a weak microwaveradiation that can in principle be used to probe the current phase relation. Forexample, in the no-Majorana state a junction biased at 4μV will emit microwaveradiation at 2GHz, and in the Majorana state it will emit radiation at 1GHz. Itis not yet clear what the best detection method is, but we propose that futuristictunable CNT based JPAs can perhaps be a candidate, as they are in principlecompatible with the needed magnetic fields.

Although it remains to be seen if CNT junctions can be used to fabricate usefultransmon qubits, we propose one futuristic experiment in which the high magneticfield compatibility is used to full advantage.

Hybrid P1-center/transmon CNT qubit compatible with large in-planemagnetic fieldThe main source of decoherence of the electrons spin of a diamond NV centeris determined by the environment spin bath. The spin bath can be formed byelectron spins associated with single nitrogen atoms (P1 centers) or the nuclearspin of 13C atoms in diamond [54]. NV center coherence times can be increased bypolarizing the P1 centers in a magnetic field [55]. The P1 spin bath itself can beused in quantum information processing, for instance as a quantum memory (spinsystems in diamond have orders of magnitude longer coherence times comparedto those of superconducting qubits). An important advantage of P1 centers overNV centers is that they are easily found in grown diamond, while NV centersappear in low densities in such samples.

In contrast to NV centers in diamond, P1 centers are insensitive to optical fields,but can be addressed by microwave fields. However, a large magnetic field isneeded to get transitions of P1 states in the microwave regime (B ≈ 0.3T toget 7GHz transitions). In such fields, the polarization of a P1 electron spinensemble can be studied by coupling them strongly to a superconducting LCresonator [56]. Superconducting devices need to be compatible with such fieldsin these experiments, which was achieved using high Tc NbTiN films, with acritical field of B‖ � 0.3T.

In the recent experiment by Ranjan et al. [56], the P1 spin resonance was tunedin resonance with the LC resonator by a change of the ∼ 300mT magnetic field.This sometimes resulted in jumps in the LC resonance frequency, likely due tothe rearrangement of trapped flux in the resonator. One way to circumvent thisproblem is to keep the field constant, and add a SQUID in the resonator. In thisway the resonance frequency can be tuned by an additional small field changingthe flux threading the SQUID [19]. We suggest that CNT SQUIDs can potentiallybe used for this purpose, as they in principle can operate at B‖ ≈ 0.3T fields.

In a more futuristic device (see Fig. 7.8), CNTs are used to make a transmonqubit that is compatible with such fields as well, and a similar experiment asdone by Kubo et al. could be pursued. In such an experiment the P1 ensemblecould be used as a quantum memory, to store quantum information manipulatedby the transmon qubit.

179


VgVg VgVg

B ,1

B ,2

P1

Figure 7.8: Hybrid system to couple P1-centers in diamond to a CNT basedtransmon qubit. CNT based Josephson junctions can carry a sizable supercurrentup to B ≈ 0.5T (see Fig. 5.31). They can possibly be used as Josephson element intransmon qubits, making them compatible with in-plane magnetic fields. At B‖ = 0.3Tit is convenient to study P1 centers in diamond. CNT based transmon qubits that workat such fields could be coupled to P1 centers in a tunable hybrid device as is shownhere. Tunability of the LC resonance is achieved by an embedded CNT SQUID. Figureadapted from Kubo et al. [19].

7.6 Consideration of future directions

In this chapter I have given an overview of many (speculative) experiments thatcould possibly be done using the new technology that we have developed. Ofcourse the main question is, which of these experiments should be pursued, andwhich one will make the largest impact, in say, ten years from now? In thisfinal section of my thesis I will give a personal consideration on the future ofsuperconducting suspended CNT Josephson junctions.

Carbon nanotubes distinguish themselves from (all) other materials because theyare the only truly one-dimensional wires and show ballistic transport. Theyalso have small mass. In my opinion these properties make CNTs unique, andexperiments in which they can be exploited to the full extend cannot be donein any other system. I will now comment on future directions in which theseproperties of CNTs can be used.

CNTs as mass sensorsAs we have seen in this thesis, their small mass makes CNTs interesting for massand force sensing experiments. Now that CNT resonators have already been madethat can weigh a single proton [57], I foresee a CNT based electronic-nose devicewith receptors that are chemically bonded to the CNT and can specifically “sniff”molecules. This is a practical application, but since the proof of principle hasalready been shown, it is to me more interesting from an engineering viewpointthan from a physics point of view.

180

7.6. Consideration of future directions

CNT motion in the quantum regimeWe have discussed that the zero point motion is the ultimate limit for masssensing, and I think that the effort to reach this limit now, is mainly motivatedto study the quantum nature of a macroscopic CNT resonator. But as wasdiscussed in the first section, it is at this point not clear how this can be done bycoupling a CNT resonator to a transmon qubit directly. For this reason I wouldnot pursue direct coupling of a CNT resonator to a transmon qubit.

I think it will be useful to embed CNTs as Josephson elements in transmon qubitsand other superconducting devices, because their gate tunability is potentiallyuseful (to tune the qubit gap for example) but more importantly: They will allowoperation at elevated magnetic fields. Until now experiments on superconduct-ing qubits were limited to low magnetic fields, and I expect that some interestingstudies of decoherence could be done as function of magnetic field, perhaps thecoherence time of qubits increases if nuclear spins in for example the tunnel-barriers or qubit environment can be omitted or polarized. Using CNTs has anadditional advantage, in principle the Josephson energy of a qubit junction canbe increased by a factor of ten, just by increasing the number of CNTs formingthe junction by a number of ten as well. I see no reason why the in-plane criticalfield of such devices would be much different. Apart from the pick-and-placestamping fabrication method, this is another advantage of CNT junctions overfor example the diamond SQUID-type junctions.

In my opinion effort should be made to inductively couple the motion of a sus-pended CNT Josephson junction to an LC resonator as we presented in Sec. 7.3.I think the numbers are encouraging and that in this experiment possibly the res-onator could be cooled to the ground state. It can also be the first step towardsa hybrid system with an LC cavity, a transmon qubit and a CNT mechanicalresonator (enabling experiments as in Ref. [4]). In any case in such a systemthe inductance and current-phase relation of a CNT can be studied, which isinteresting in itself. Also I would encourage to develop a Josephson parametricamplifier to detect CNT vibrations.

Josephson parametric amplifiers based on CNTsTunable Josephson parametric amplifiers have already demonstrated their hugepotential by measurement of the ground state motion of mechanical resonators.Also the operation of these amplifiers is restricted to small magnetic fields. Again,if all Josephson elements could be replaced with carbon nanotubes I foresee thatnew applications of JPAs will show up in which they can be used at high magneticfields.

I think that this can have a large impact, and the applications of such devices canalso extend beyond the field of nanomechanics. They could for example be usedas a new tool to study properties of magnetic molecules, or measure Josephsonradiation emitted by Majorana junctions.

181

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186

Appendix A

Additional data

In this appendix we present additional data on the graphene and carbon nanotubeexperiments presented in Chapters 4 and 5. To prevent distraction from the mainstorylines we omitted this data there. The data presented here has been referredto in the main text.

A.1 Additional data on graphene

A.1.1 Graphene IswRn product

We have studied four Nb junctions, six Ti/Nb/Ti junctions and five Ti/V/Tijunctions. In Fig. A.1 we give an overview of the resistance and switching currentof our junctions. We find that the IswRn product is enhanced in Ti/Nb/Tijunctions compared to only- Nb junctions, and it is the largest in our Ti/V/Tijunctions. Contact spacing (∼ 200 nm) and graphene preparation was similar inall batches.

0 0.75 1.5Nb

Ti/Nb/Ti

Ti/V/Ti

R(k )0 0.3

Isw(mA)0 50

IswRn(mV)

.915

(a) (b) (c)

Figure A.1: Distribution of IswRn for different contacts. (a) Scatter plot ofnormal state resistance of Ti/V/Ti, Ti/Nb/Ti and Nb devices. (b) As in (a), but nowfor Isw. (c) As in (a), but now for IswRn.

187

A. Additional data

A.2 Additional data on carbon nanotubes

A.2.1 Large gate range characterization

In this section we will present additional data on the characterization of our CNTdevices. We have studied the gate dependence of the mechanical resonance andhave done voltage and current bias spectroscopy. We find that our devices cancarry a supercurrent across the whole (conducting) gate range, and have tunablemechanical modes in the fr ∼ 1GHz regime. The absence of modes at f < fr(data not shown) is an indication that we study the fundamental mode, but wedo not know this for sure.

Mechanical characterizationA suspended CNT can be pulled closer to the gate by the static gate force.Thereby its resonance frequency can be tuned [1, 2]. The static gate force isproduced by the DC gate voltage V0 in the following way (see App. C):

Fstatic =∂Cg

∂y

V 20

2. (A.1)

In the absence of additional (static) forces a parabolic dependence of the resonatorfrequency on the gate voltage is expected. We have observed such a dependencein device one and device two, which is shown in panel (a) of Figs. A.2 and A.3.

Electrical characterizationWe have performed voltage bias spectroscopy in a two-point measurement con-figuration. Such measurement is useful because it allows us to obtain an approxi-mate value of the conductance. Extracting the absolute value of the conductancefrom such measurement is non-trivial, because the conductance of the device ishighly non-linear and its resistance has the same order of magnitude as the se-ries resistance in the line (device resistance: R ∼ 10 kΩ, circuit series resistance:R ∼ 8 kΩ). This implies that the voltage drop across the device depends heavilyon its conductance, and is a fraction of the voltage at which the voltage sourceis set. The conductance plots in Fig. 5.5 are extracted from a dataset obtainedby a four point (current biased) measurement.

In panel (b) in Figs. A.2 and A.3 we show a colorplot of the (approximate) conduc-tance. We recognize the typical checkerboard pattern of ballistic, open quantumdots across almost the full gate range (see Sec. 2.1.3). The CNT bandgap is visibleclose to Vg = 0. We have used these datasets to extract the bandgap presentedin Tab. 5.1. This is possible because close to the bandgap the full applied voltagedrops across the device. On the few-electron side we recognize faint Coulombdiamonds. We have used these to extract the charging energy, gate capacitance,and the total capacitance of the CNT quantum dot in the few-electron regime [3].In Tab. A.1 we plot the extracted values. Here eΔVsd is the height of one of thefirst Coulomb diamonds, and eΔVg is its width. We have used α = ΔVsd/ΔVg,Cg = e/ΔVg, and CΣ = e2/ΔE, where ΔE = eΔVsd.

Usually the the gate capacitance is estimated by a model in which the CNT ismodeled as a cylinder suspended above an infinite plate (see Eq. (B.2) and [4]).

188

A.2. Additional data on carbon nanotubes

Table A.1: Few-electron quantum dot charging energy and capacitance.

eΔVsd eΔVg α Cg Cg,model CΣ

(meV) (meV) (10−3) (aF) (aF) (aF)1 8.2 309 30 0.5 4.6 19.52 21 252 80 0.6 8.3 7.6

Deviations from this model occur when the distance to the gate is smaller or onthe order of the length of the CNT. Also screening of the electric field by thecontacts is not accounted for. Typically these effects result in an overestimationof the gate capacitance [5], which agrees with our findings. The capacitanceCΣ = Cg+Cs+Cd is the total capacitance to the gate, source and drain contacts.We find that the capacitance to the source and drain contacts is dominant, whichis also indicated by the small alpha factor. We note that the alpha factor is anorder of magnitude larger on the electron side compared to the hole side (seeTab. 5.1) which is probably due to a different confinement of electrons and holes.

In panel (c) in Figs. A.2 and A.3 we show the current bias spectroscopy. Thesupercurrent branch is clearly visible at I ∼ 0 nA. We typically observe multiplesteps in an IV trace, which is especially apparent in Fig. A.2c and its inset. Theheight of these steps are not tunable by gate or temperature (data not shown).They are probably Fiske resonances, which are like Shapiro steps, but are due tohigh frequency electrical LC modes in the device circuit [6].

189

A. Additional data

-2

0

2

900

910

920

930 d2I/dfdVg(a.u.)

204060

-13.5 -9 -4.5 0 4.5 9 13.5

-30

-15

0

15

30

Vg(V)

V(m

V)

dI/dV(μS)

dV/dI(kΩ)

0

50

100

-5

0

5

I(nA

)f(

MH

z)

(a)

(b)

(c)

-5 0 5-50

0

50

V(μ

V)

I(nA)

Figure A.2: Large gate range characterization, device one. (a) Mechanicalresonance as a function of gate voltage. Vsd ∼ 750μV, the current through the deviceis I ∼ 50 nA, P = −20 dBm. The faint green line is a parabolic guide to the eye. (b)Voltage bias spectroscopy, T ∼ 80mK. Conductance values are approximate, becausethe voltage drop across the device is not well known. (c) Current bias spectroscopy.The supercurrent branch is the zero resistance region close to I ∼ 0 nA. The dashedline is at Vg = −8.715V, and the inset is a linecut from the data at this point.

190


1285

1305

1325

f(M

Hz)

204060

-6 -4 -2 0 2 4 6

-30

-15

0

15

30

Vg(V)

V(m

V)

dI/dV( S)

V/Vmax(a.u.)

-0.5

0

0.5

I(nA

)

-2

-1

0

1

2

-2 0 2-15

0

15

V(

V)

I(nA)

d2I/dfdVg(a.u.)

(a)

(b)

(c)

Figure A.3: Large gate range characterization, device two. (a) Mechanicalresonance as a function of gate voltage. Vsd ∼ 1mV, the current through the device isI ∼ 80 nA, RF power is P = −15 dBm and T ∼ 150mK. (b) Voltage bias spectroscopy,T ∼ 2K (measured in dipstick setup). Conductance values are approximate, becausethe voltage drop across the device is not well known. (c) Current bias spectroscopy.The supercurrent branch is the zero voltage region close to I ∼ 0 nA. The dashed lineis at Vg = −5.04V, and the inset is a linecut from the data at this point. Due to atechnical problem with the amplifier, the voltage noise was large in this measurement.That is why we have stretched to colorscale for each line to obtain good contrast.

191

A. Additional data

A.2.2 Effect of dV/dI on signal amplitude

The amplitude of the measured voltage ΔV is heavily dependent on the slopedV/dI of the IV curve at the bias point. At a current bias of 7 nA, one orderof magnitude above the switching current, the signal is still observed, but itsbias dependence is much weaker because of the absence of Shapiro steps at thisbias. This is shown in Fig. A.4. Because of this reason we have chosen this point(indicated by a star) as a working point for many of our measurements. Thisallows us to extract dependence of ΔV on other parameters (notably magneticfield and power), at low bias the size of the signal ΔV is dominated by changesof dV/dI that also occur as a function of these parameters.

-0.3 0 0.3I(nA)

V(

V)

-2

0

2

V(

V)

f fr

(a)

f fr

0

0.2

0.4

-0.3 0 0.3I(nA)

(c)

6.7 7 7.30

0.2

0.4

I(nA)

V(

V)

46

48

50

V(

V)

(b)

6.7 7 7.3I(nA)

(d)

Figure A.4: Effect of dV/dI on signal amplitude. (a) Two IV s measured on/offresonance, at low current bias where there are (weakly visible) Shapiro steps. (b) Asin (a), but now at high current bias. There are no Shapiro steps here. The star is alsoindicated in Fig. 5.10, and indicates a typical operation point. (c) The amplitude ofthe resonance signal ΔV is proportional to dV/dI, and oscillates as a function of bias.(d) As in (c), but now at high bias. Because there are no Shapiro steps, ΔV (I) isalmost constant here. All data was taken at Vg = −8.715V, with f = 919.14MHz andP = −39 dBm.

192


A.2.3 Detailed field dependence device two

In Fig. 5.27 we presented the detailed field dependence on device one. Here wewill do so for device two in Fig. A.5.

In device two we observe oscillations of Im/(dG/dVg) on top of a gradual decreaseof the signal amplitude up to B = 2.4T. Notice that in this device the signchange occurs at B ∼ 2.5T. In device two there are oscillations in ΔV as wellas in dG/dV . When correcting for the oscillations in dG/dVg, the oscillationsstill persist in Im/(dG/dVg). For comparison, we have also plotted the fielddependence of the switching current Isw in panel (d).

0 0.9 1.8 2.40

0.25

0.5

B(T)

B(T)

B(T)

B(T)

0

10

20

0 0.9 1.8

0 0.9 1.80

dG/d

Vg(

S/V

)dV

/dI(

h/e2

)

V(n

V)

y(

AV

/S)

(a) (b)

(c) (d)

Im (pA)

Isw (nA)

Bc=2.1T

0 0.9 1.8

-300

-150

0

0.5

1

0

0.05

0.1

0.15

0

75

150

Figure A.5: Overview of resistance, transconductance and signal amplitudeas a function of field, device two. (a) Resistance dV/dI as a function of field, inunits of h/e2 = 25.8 kΩ. At Bc = 2.1T a small jump of ∼ 0.86 kΩ indicates the criticalfield of the contacts. (b) Signal size ΔV . Mixing current ΔIm is defined as ΔV/(dV/dI).(c) Transconductance as a function of field. Transconductance is reduced by a factorof two between B = 0 and B ∼ 0.8T. (d) By dividing the mixing current by thetransconductance we extract a signal Δy ·κ that is proportional to the CNT amplitude.Data taken at Vg = −5.025V and I = 7nA.

193

A. Additional data

A.2.4 Bias dependence corrected for ΔV/ΔI

In this section we present the size of the mixing current ΔI, which is the mea-sured voltage signal ΔV , divided by the (off-resonance) resistance dV/dI, ΔI =ΔV/(dV/dI). We present ΔI(I), and study its dependence on magnetic field.We have measured the signal amplitude, as is shown in Figs. 5.29-5.30. Thisdata is plotted again in panels (a) of Figs. A.6 and A.7.

0

0.2

0.4

I(nA)

V(

V)

(a) (b)

-70 0 706

7.25

8.5

I(nA)

dV/d

I(k

)

-70 0 70

0T

0.31T0.61T

0

35

70

I(nA)-70 0 70

I(pA

)

(c)

0T

0.31T0.61T

0.61T

0.31T

0T

Figure A.6: Bias dependence, device one. (a) Voltage amplitude of mechanicalresonance signal ΔV , extracted from a Fano-lineshape fit to the measured voltage, asa function of current bias and magnetic field. (b) Resistance dV/dI as a functionof current bias and field. (c) Current amplitude of mechanical resonance signal ΔI,extracted from data in panels (a) and (b).

0

75

150

I(nA)

V(n

V)

(a) (b)

-90 0 905

7.5

10

I(nA)

dV/d

I(k

)

-90 0 90

0T

1.8T2.4T

0

10

20

I(nA)-90 0 90

I(pA

)

(c)

2.4T

1.8T

0T

0T

1.8T2.4T

Figure A.7: Bias dependence, device two. (a) Voltage amplitude of mechanicalresonance signal ΔV , extracted from a Fano-lineshape fit to the measured voltage, asa function of current bias and magnetic field. (b) Resistance dV/dI as a functionof current bias and field. (c) Current amplitude of mechanical resonance signal ΔI,extracted from data in panels (a) and (b).

The disappearance of the signal enhancement at I ∼ 0 in the SC regime cannotonly be accounted for by the increased dV/dI in this regime (panels (b)). Inpanels (c) we plotted the signal amplitude in current.

194


A.2.5 Order of magnitude estimation of transduction en-hancement

In this section we present the order of magnitude estimation for the transductionenhancement in the superconducting regime vs. the normal regime. Here wepresent data on resistance- and transconductance dependence on power.

Transduction coefficient ΔV/Vrf

We calculate the transduction coefficient ΔV/Vrf by dividing the measured ΔVby the applied Vrf . This is plotted in Fig. A.8.

0 0.25 0.50

0.6

1.2

Vrf(mV)

V/V

rf(1

0-3 )

(a)

B=0T

0 1 2 3 4 50

6

12

Vrf(mV)

V/V

rf(1

0-6 )

(b)

B=2.4T

0 1 2 30

0.2

0.4

Vrf(mV)

V/V

rf(1

0-3 )

0 1 2 30

0.04

0.08

Vrf(mV)

V/V

rf(1

0-3 )

(c) (d)

2.4T

0T

2.4T

Figure A.8: Transduction coefficient ΔV/Vrf . (a) Data from device one, measuredat B = 0T. (b) Data from device one, measured at B = 2.4T. (c) Data from devicetwo, measured at B = 0T and B = 2.4T. (d) Zoom in of (c).

Power dependence of transconductance and resistanceIn our model the transduction is mediated by the transconductance and theresistance. We plot dV/dVg (at I = 7nA) because here we are only interested inthe relative change of the transconductance. We have also measured the powerdependence of the local slope dV/dI at I = 7nA. This data is presented inFigs. A.9-A.10.

195

A. Additional data

14 24 34 440

0.25

0.5

Vrf( V)

dV/d

I(h/e

2 )

0 1 2 3 4 50

0.25

0.5

Vrf(mV)

dV/d

I(h/e

2 )

RB=0T =6.9k

1 3 50

0.25

0.5

Vrf(mV)

dV/d

I(h/e

2 )

0 1 2 3 4 50

30

60

90

Vrf(mV)

|dV

/dV g

|(V

/V)

14 24 34 440

30

60

90

Vrf( V)

|dV

/dV g

|(V

/V)

1 3 50

30

60

90

Vrf(mV)

|dV

/dV g

|(V

/V)

B=0TB=2.4T

dV/dVg,B=0T = -42 V/V

RB=2.4T =8.9k dV/dVg,B=2.4T = -42 V/V

(a)

(c)

(e)

(b)

(d)

(f)

Figure A.9: Power dependence of transconductance and resistance, deviceone. (a,c,e) Power dependence of dV/dI, measured at B = 0T and B = 2.4T. (b,d,f)Power dependence of dV/dVg, measured at B = 0T and B = 2.4T.

196


0 0.25 0.50

0.25

0.5

Vrf(mV)

dV/d

I(h/

e2)

0 1 2 30

0.25

0.5

Vrf(mV)

dV/d

I(h/

e2)

0.5 1 1.50

0.25

0.5

Vrf(mV)

dV/d

I(h/

e2)

0 1 2 30

0.15

0.3

Vrf(mV)

|dV

/dV

g|(m

V/V

)

0 0.25 0.50

0.15

0.3

Vrf(mV)

|dV

/dV

g|(m

V/V

)

dV/dVg,B=0T =-0.30mV/VdV/dVg,B=2.4T =-0.17mV/V

0.5 1 1.50

0.15

0.3

Vrf(mV)

|dV

/dV

g|(m

V/V

)

(a)

(c)

(e)

(b)

(d)

(f)

B=0TB=2.4T

RB=0T =9.1kRB=2.4T = 11.2k

Figure A.10: Power dependence of transconductance and resistance, devicetwo. (a,c,e) Power dependence of dV/dI, measured at B = 0T and B = 2.4T. (b,d,f)Power dependence of dV/dVg, measured at B = 0T and B = 2.4T.

197

A. Additional data

Order of magnitude estimation of transductionHere we will give the order of magnitude estimation of the transduction. Forsimplicity we will assume that the resonator is driven beyond Fc (see Eq. (2.46)),and that its amplitude is constant as a function of power. We will compare signalsat B = 0 (SC regime) and B = 2.4T (N regime).

Device oneIn the SC regime: ΔVSC/Vrf ∼ 10−3, and in the N regime: ΔVN/Vrf ∼ 10−5.The transduction enhancement in the SC regime is on the order of 102. Thetransduction enhancement due to resistance change is 6.9/8.9 ∼ 0.75, and due totransconductance is 82/42 ∼ 2 (values extracted from Fig. A.9). If we correct forresistance and transconductance change we find 102/(2 · 0.75) ∼ 60 ∼ 102. Thisimplies there is about two orders of magnitude mixing enhancement in the SCregime compared to the N regime in device one.

Device twoIn the SC regime: ΔVSC/Vrf ∼ 0.4 · 10−3, and in the N regime: ΔVN/Vrf ∼0.05 · 10−3. The transduction enhancement in the SC regime is on the order of10. The transduction enhancement due to resistance change is 9.1/11.2 ∼ 0.8, anddue to transconductance is 30/17 ∼ 1.8 (values extracted from Fig. A.10). If wecorrect for resistance and transconductance change we find 8/(1.8 ·0.8) ∼ 6 ∼ 10.In device two we find about one order of magnitude mixing enhancement.

198


A.2.6 Magnetic field dependence of Q and fr

In this section we present the field dependence of quality factor and linewidth.These are fit parameters extracted from the datasets in Figs. 5.22 and 5.25.

In device one (Fig. A.11) we observe a strong frequency dependence that is cor-related with the size of the signal. The resonance frequency is tunable withfr = 919.165±0.01MHz. We also observe a higher quality factor when the signalamplitude is small. Because the accuracy of the Q-factor fit goes down whenthe signal amplitude is reduced, it is hard to make a strong conclusion here.We find that a reduction of signal (around B = B0−π and towards B = Bc) isaccompanied with an increasing Q-factor.

0.15

0.3

Q(1

06)

919.165

f r(M

Hz)

I=7nAI=-7nA

B(T)0 0.7 1.4 2.1

B0-0

919.155

919.175

Figure A.11: Magnetic field dependence of Q and fr, device one. Toppanel: fr(B), bottom panel: Q(B). Fit parameters Q and fr are extracted from Fano-lineshapes fitted to the datasets presented in Fig. 5.22.

In device two (Fig. A.12) the situation is different, notably the resonance fre-quency and quality factor drop (rather than peak, as is the case in device one)close to B0−π = 1.94T. We have no model to account for these results, furtherexperiments could be done to investigate this in detail.

1.88 1.94 20

15

30

Q(1

03)

f r(M

Hz)

1332.91

1332.96

1332.86

1332.93

0

50

100

Q(1

03)

f r(M

Hz)

(a)

0 1.5 3 4.5B(T)

1332.88

1332.98

B(T)

(b)

B0-B0-

Figure A.12: Magnetic field dependence of Q and fr, device two. (a,b) Toppanels: fr(B), bottom panels: Q(B). Fit parameters Q and fr are extracted fromFano-lineshapes fitted to the datasets presented in Fig. 5.25.

199

Bibliography

Bibliography



[3] L. Kouwenhoven, D. Austing, and S. Tarucha, Few-electron quantum dots,Reports on Progress in Physics 64, 701 (2001).

[4] R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and P. Avouris, Single- andmulti-wall carbon nanotube field-effect transistors, Applied Physics Letters73(17), 2447–2449 (1998).

[5] W. Lu, J. Xiang, B. P. Timko, Y. Wu, and C. M. Lieber, One-dimensional holegas in germanium/silicon nanowire heterostructures, Proceedings of the Na-tional Academy of Sciences of the United States of America 102(29), 10046–10051 (2005).


200

Appendix B

Vibrating carbon nanotubequantum dots

It is possible to capacitively read-out the motion of a CNT above a gate, using thegate dependence of the conductance. The purpose of this chapter is to analyze andsimulate the effect of motion, oscillating gate voltages and oscillating bias voltageson the conductance. In particular we are interested in the CNT mechanicalresonance lineshapes and its dependence on the transconductance, ∂G/∂Vg. Inexperiments we find certain lineshapes and gate dependence. This work helpsus to interpret the origin of the experimentally found signals, for example todistinguish between mixing or rectification as the dominant mechanism.

In Sec. B.1 we present our analysis of the effect of oscillating gate voltage anddisplacement on the CNT conductance. Here we assume that there is no oscil-lating bias voltage across the device. We find that in this case we can extractLorentzian lineshapes due to a rectification of the motion and Fano lineshapesdue to mixing of the motion with oscillating gate voltages. Also we make anestimation on the dominant mechanism for realistic devices, that depends on thequality factor of the resonator.

In Sec. B.2 we present a numerical analysis where we consider the effect of oscil-lating gate voltage and oscillating bias voltage on the conductance of a quantumdot. In this analysis we do not consider mechanical motion, but as will be dis-cussed in Sec. B.1, since an oscillating CNT is analogous to an oscillating gatevoltage, it can also be understood as an analysis of rectification and mixing inthe presence of mechanical motion and an oscillating bias voltage.

B.1 Effects on conductance by displacement

B.1.1 Model

We consider a suspended CNT, connected to source and drain (Fig. B.1). TheCNT has a capacitance Cg to a gate. An RF antenna is capacitively coupled to

201

B. Vibrating carbon nanotube quantum dots

the CNT and gate, and can be used to capacitively induce a voltage on the gate.Here we analyze the effect of an oscillating gate voltage, and displacement of theCNT on its charge and conductance.

ydrain

gate

sourcey

L

CNT

VG + VG

Figure B.1: Suspended CNT above a gate. The CNT is resting on a source anddrain contact which can be used to measure its conductance.

As we will see, the effect of displacement and oscillating gate voltage is an effectivechange of the charge on the nanotube and hence its conductance, at points ingate space where the conductance is changing a lot as a function of the charge.This is the case at Coulomb peaks. This is summarized in Fig. B.2.

0

G

q(VG)

q³VG, y

´

V0

q=N q=N+1

Figure B.2: Effect of oscillating charge on conductance. Charge on the CNTcan be changed by applying a gate voltage. Coulomb peaks appear between two stablecharge configurations. There the conductance is very sensitive to changes in (average)charge. These can be induced by an oscillating gate voltage or a moving CNT.

The effect of displacement and oscillating gate voltages on the CNT conductanceis best analyzed by considering the charge on the CNT:

q = CgVg . (B.1)

In a (closed) CNT quantum dot the charge is quantized in stable regimes betweenCoulomb oscillations. On a Coulomb peak the average charge on the dot isallowed to change by fractions of e. As we will see, displacement of the CNTcauses a change in the capacitance of the CNT to the gate, which modifies thecharge, when the gate voltage has been tuned on a Coulomb peak. The CNTis normally modeled by a thin metallic wire above an infinite plate [1]. In thismodel the capacitance is given by:

Cg =2πε0εrL

ln (2y/r). (B.2)

202

B.1. Effects on conductance by displacement

Here y is the distance between the CNT and the gate, r is the CNT diameter andL is the CNT length. This model can be used when y � r and L > y [2, 3]. ForL < y the capacitance is generally smaller than predicted by the simple model,because of screening by the metallic leads [3]. This effect starts when L ∼ y andis not accounted for in our analysis.

The charge on the dot can be changed with Cg or Vg. A nearby RF antenna

can capacitively induce a modulation Vg of the gate voltage, Vg = V0 + Vg. HereV0 is the static gate voltage. When the RF signal is tuned to a frequency onresonance with one of the mechanical modes of the CNT resonator, displacementcauses a modulation of the capacitance Cg. On resonance y = y0 + y, where y0is the static distance to the gate and y is the displacement. Next we derive thedependence of Cg on y.

In typical devices y/r is on the order of 102 . . . 103. In this regime ∂Cg/∂y ≈constant for small displacements y � y. Using Eq. (B.2) we write (using L ∼ y):

∂Cg

∂y≈ − 1

ln (2y/r)

Cg

y. (B.3)

Experimental devices used in this thesis are typically in the regime where y/r ≈300, in this case we write:

Cg ≈ −∂Cg

∂yln(600)y , (B.4)

Cg ≈ 6αy . (B.5)

(B.6)

Where we have defined α ≡ −∂Cg

∂y, which is determined by the geometry of the

device. Equation (B.4) implies Cg ∝ y. Displacement of the CNT is transduced

to a change in capacitance, Cg = C0+Cg. As we will see later on this gives rise toa change in conductance, which can be used to read-out the CNT displacement.First we consider the effect without displacement.

In both situations we consider there is a voltage Vg = V0+ Vg applied to the gate.

We assume that Vg = a sin(ωt+ ϕ).

B.1.2 Effect of AC signal without CNT displacement

We start with a fixed capacitance between the CNT and the gate:

Cg = C0 . (B.7)

A change of the gate voltage Vg results in a change of charge q on the CNT,which results in a change of conductance. Now we expand the conductance forsmall changes in gate voltage:

G(Vg + Vg

)= G0 +

∂G

∂VgVg +

∂2G

∂V 2g

V 2g +O

(V 3g

)(B.8)

203


After time integration only the terms with even powers in Vg contribute to theconductance.⟨

G(Vg + Vg)⟩= G0 +

∂2G

∂V 2g

⟨V 2g

⟩+O

(⟨V 4g

⟩)(B.9)

This effect results in a change of conductance at points where ∂2G/∂V 2g is large.

B.1.3 Effect of AC signal with CNT displacement

On resonance the change of conductance due to displacement of the nanotube alsohas to be taken into account. The capacitance of the CNT to the gate dependson the displacement.

Cg = C0 + Cg,Cg � Cg0 (B.10)

q =(C0 + Cg

)(V0 + Vg

)= (B.11)

C0V0 + C0Vg + CgV0 + CgVg (B.12)

Now not only a change of gate voltage Vg results in a change of charge q, but also

a change of capacitance Cg can change the charge, and hence the conductance.

Because as we have seen in Eq. (B.4) that Cg ∝ y this means we have to expandthe conductance for small changes in gate voltage as well as displacement.

G(Vg + Vg, y + y

)= G0+

∂G

∂VgVg+

∂G

∂yy+

1

2

∂2G

∂V 2g

V 2g +

1

2

∂2G

∂y2y2+

1

2

∂2G

∂Vg∂yVgy+h.o.

(B.13)

With h.o. we denote higher-order terms. After time integration terms with evenpowers in y or Vg and the term with Vgy contributes to the conductance:

⟨G(Vg + Vg, y + y

)⟩= G0+

1

2

∂2G

∂V 2g

⟨V 2g

⟩+1

2

∂2G

∂y2⟨y2⟩+1

2

∂2G

∂Vg∂y

⟨Vgy

⟩+h.o.

(B.14)

Only the last two terms contribute to a change of conductance due to displace-ment. The contribution from the term with

⟨y2⟩is different from that of the

term with⟨Vgy

⟩. As we will demonstrate next, a phase difference ϕ between Vg

and y results in a conductance change that is different from the term with y2.

In Sec. 2.4 we derived an expression for the complex amplitude yω. Due to theDuffing nonlinearity this is an implicit equation. For simplicity and because itdoes not change the picture we will show here, in the following we consider theDuffing term to be zero, β = 0.

yω =Fω

m

1

2ω0ν − iΓω0(B.15)

By inverse Fourier transform the amplitude of the motion can be resolved (seeSec. 2.4). The time average of the change of conductance on resonance can befound directly from the complex amplitude. Next we consider this response foreach contributing term.

204

B.1. Effects on conductance by displacement

Term with⟨y2⟩

Multiplication in the time domain corresponds to a convolution in the frequencydomain:⟨

y2⟩= Re [yω ∗ yω] (B.16)

The response of the conductance to the term with⟨y2⟩has a Lorentzian lineshape

as a function of detuning and is plotted in Fig. B.3.

Term with⟨Vgy

⟩In this term we have to account for the phase difference ϕ between Vg and y. In

the frequency domain this is equal to multiplication with eiϕ (assuming Vg is adelta function in the frequency domain), which results in:

〈y (t, ϕ)〉 = Re [yω] cos(ϕ)− Im [yω] sin(ϕ) (B.17)

The response of the conductance to the term with 〈y(t, ϕ)〉 has a phase-dependentlineshape as a function of detuning (Fig. B.3). In this case, lineshapes can beasymmetric as well as Lorentzian. The asymmetric lineshapes can be fitted to aFano function (see Eq. (2.62)), and are referred to as Fano lineshapes [4].

-0.2 0 0.20

1

(a.u.)

y2®

(a)

-0.2 0 0.2

0

1

/20- /2

(a.u.)

D ˜ V GyE (b)

Figure B.3: Lorentzian and Fano lineshapes. (a) Response of conductance from

term with⟨y2⟩. (b) Response of conductance from term with

⟨Vgy

⟩

.

When an asymmetric lineshape is observed, it is possible that the term with⟨Vgy

⟩is larger than that with

⟨y2⟩. By comparison of the magnitude of both

terms we can find out when one dominates the other:

|Lorentzian/Fano| = |(1

2

∂2G

∂y2⟨y2⟩)

/

(1

2

∂2G

∂Vg∂y

⟨Vgy

⟩)| (B.18)

As we will show, the r.h.s. can be expressed in terms of the derivative of con-ductance to charge. This is useful because these derivatives will cancel and|Lorentzian/Fano| is found. Next we express the Lorentzian and Fano term interms of the derivative of conductance to charge.

205


Lorentzian term

q =∂Cg

∂yyVg = αVgy (B.19)

∂2G

∂y2⟨y2⟩=

∂2G

∂q2⟨q2⟩

(B.20)

∂2G

∂q2⟨q2⟩=

∂2G

∂q2

(∂Cg

∂yVg

)2 ⟨y2⟩

(B.21)

∂2G

∂q2=

∂2G

∂ (CgVg)2 =

∂2G

∂V 2g

1

C2g

(B.22)

∂2G

∂y2⟨y2⟩=

∂2G

∂V 2g

(Vg

Cg

∂Cg

∂y

)2 ⟨y2⟩

(B.23)

Where we have assumed that the capacitance does not change with Vg. Now wecan write:

〈G (y, Vg)〉 = 1

2

∂2G

∂V 2g

(Vg

Cg

∂Cg

∂y

)2 ⟨y2⟩

(B.24)

This is the expression for the sensitivity of the conductance to 〈y2〉 [5].Fano term

1

2

∂2G

∂Vg∂y

⟨Vgy

⟩=

∂2G

∂q2VgCg

∂Cg

∂y

⟨Vgy

⟩(B.25)

Now we have expressed both terms in ∂2G/∂q2 and we can estimate |Lorentzian/Fano|in terms of the resonator properties. Substitution of Eq. (B.3) gives:

|Lorentzian/Fano| = 1

ln (2y/r)

Vg

y

y

Vg

. (B.26)

Substitution of Eqs. (2.43), (2.48), (2.50), (2.51) and (2.52) in Eq. (B.26) gives:

|Lorentzian/Fano| = 1

2

(2π

22.4 ln (2y/r)

)2

Q . (B.27)

Where we have taken y ≈ L.

In a typical device y/r ≈ 300, which gives the simple expression:

|Lorentzian/Fano| ≈ 1

1000Q . (B.28)

From this estimation we learn that for Q � 1000 the response of the CNTresonator in the change of conductance is Lorentzian. For Q � 1000 the responsecan have a Fano lineshape. A physical interpretation of this result is as follows:

When y/y � Vg/Vg the mixing term⟨Vgy

⟩is very small compared to

⟨y2⟩.

Hence the change of conductance for resonators which have a big amplitudeon resonance has a smaller component from mixing compared to a resonator

206

B.2. Conductance through a QD in the presence of oscillatingvoltages

with a small amplitude, for a given Vg. The proportionality factor 1/1000 is aconsequence of the device properties.

In experiments with similar devices as discussed above, Fano lineshapes are usu-ally found with relatively low Q-factors � 1000: 56 [6], 80 [7], 200 [8], 1665 [9],(high Q, but not a CNT: Q = 104 [10]). It has to be noticed that in these ex-periments a mixing technique (oscillating bias voltage) is used to read out themotion of the nanotube, that can also give a Fano lineshape.

Lorentzian lineshapes are found in devices with relatively large Q-factors: 5100[11] and 105 [5]. In these experiments a rectification technique is used to readout the motion of the nanotube.

B.1.4 Conclusion

When the position of the CNT is fixed, an oscillating gate voltage results in achange of the DC conductance. The maximum difference appears where ∂2G/∂V 2

g

is large. When the frequency of the oscillating gate voltage is on resonance,CNT motion can be driven by an oscillating Coulomb force. This results in achange of the DC conductance, which is maximum when ∂2G/∂V 2

g is large. Tworesponse regimes can be distinguished. In the first regime the effect of the CNTdisplacement is dominating (rectification), and in the second the effect of theoscillating gate voltage is dominating (mixing). In the first case the frequencyresponse of the conductance to the displacement reflects the frequency responseof the resonator, while in the second case it is depended on the phase differencebetween Vg and y. An estimate for the crossover point is Q = 1000.

B.2 Conductance through a QD in the presenceof oscillating voltages

B.2.1 Model

The model consists of a quantum dot (QD) with a single discrete level coupled toa source and drain. The current through the QD is calculated by the numericalconvolution of the level with the bias window, as a function of the level-position(=gate voltage) and bias voltage. The QD density of states (DOS) is modeledby Lorentzian line shape with a width Γ and the source-drain DOS by a Fermi-Dirac distribution with a temperature kBT . Having AC and DC voltages on gateand/or bias, time-averaged current is resolved by numerical convolution. In theregime of interest the frequency of the AC voltage is lower than the temperature,�f < kBT . This is the classical regime where quantum effects like photon-assistedtunneling do not play a role.

The current through the QD is calculated by the convolution of the bias window

207


VGVR

L QD R

V

VL

Figure B.4: Quantum dot model showing DOS in the leads and QD. VL,R =ACvoltage on source/drain, V = DC bias voltage Vg =AC voltage on gate.

DOS with the QD level DOS.

i(E′) =∑E

(ρL(E)− ρR(E)) ρQD(E′ − E) (B.29)

The time averaged current as a function of bias and gate voltages can be found

by integration. The parameter α =(VL − VR

)/(VL + VR

)is used to describe

asymmetry of the AC voltage on the left/right lead. Three different situationsare considered: 1. VL = VR = 0, 2. VL = VR �= 0 where α = 0 and 3. VL > 0 andVR = 0 where α = 1.

B.2.2 Results

In case 1 ordinary Coulomb diamonds are resolved. In case 2 the Coulombdiamond is shifted up and down vertically by the oscillating bias voltage. Thetime average is a symmetric Coulomb diamond with smoothed edges. Noticethis case is similar to having only an oscillating gate voltage. In case 3 the ACvoltage bias is asymmetric, which produces a Coulomb diamond with asymmetricsmoothing. This is similar to shifting the Coulomb diamond up and down alongone of its edges. See Figure B.5 for the results of the numerical calculation.

From Fig. B.5d it can be clearly seen that the change in current ΔI = 〈IRFon〉 −IRFoff is the largest at the top of the Coulomb peak, where d2I/dV 2

g is maximum.In Fig. B.6, Fig. B.5d is plotted again, now with its first and second derivatives,and ΔI. It is clear that ΔI is small at the flanks of the Coulomb peak, andpeaks at the top. However in Fig. B.5f, their is a net oscillating bias across thedevice that can mix with the oscillating gate voltage and generate a time averagedcurrent that dips/peaks at the flanks of the original Coulomb peak.

B.2.3 Conclusion

The gate dependence of the DC conductance can be changed by AC voltageson bias and gate. With symmetric oscillating bias, or oscillating gate voltages,a Coulomb oscillation is smeared out symmetrically. The maximum differenceappears at points where ∂2G/∂V 2

g is large. With an asymmetric oscillating biasa Coulomb oscillation is changed in a different way, which can result in negative

208

B.2. Conductance through a QD in the presence of oscillatingvoltages

-50 0 50

-20

0

20

-50 0 50

-20

0

20

-50 0 50

-20

0

20

-50 0 50

0

1

-50 0 50

0

1

-50 0 50-2

0

2

=0

=1

VAC=0

VG (kBT)

VL-

VR

(kBT)

VG (kBT)

(a.u

.)h Ii

(a.u

.)h Ii

-1

0

1

-1

0

1

-1

0

1

(a) (b)

(c) (d)

(e) (f)

Figure B.5: Current as a function of bias and gate voltage. Figures showresults of numerical simulation in three different situations, where asymmetry of ACvoltage on the leads is varied. This asymmetry is characterized by the parameter α =(VL − VR

)/(VL + VR

). Figures on the right side are gate traces taken at small voltage

bias. Dashed line (no AC voltage) is plotted in all three figures for comparison.

differential conductance close to zero DC bias. The maximum difference appearsat points where ∂G/∂Vg is large.

209


0 10

0

I

IRF ON

d2I/dVG2

IRF ON -I

dI/dVG

VG (kBT)

curr

ent a

nd d

eriv

ativ

es (a

.u.)

=0

0 10-2

-1

0

1

I

IRF ON

d2I/dVG2

IRF ON -I

dI/dVG

VG (kBT)

curr

ent a

nd d

eriv

ativ

es (a

.u.)

=11 (a) (b)

Figure B.6: Current and derivatives as function of gate voltage and RFdrive. Left: symmetric AC bias or AC gate. Right: asymmetric AC bias. As afunction of gate voltage: current with RF on and RF off, current difference between RFon and RF off, derivatives of current with RF off. Derivatives are scaled to allow foreasy comparison.

210

Bibliography

Bibliography

[1] R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and P. Avouris, Single- andmulti-wall carbon nanotube field-effect transistors, Applied Physics Letters73(17), 2447–2449 (1998).

[2] O. Wunnicke, Gate capacitance of back-gated nanowire field-effect transis-tors, Applied Physics Letters 89(8), 083102 (2006).

[3] W. Lu, J. Xiang, B. P. Timko, Y. Wu, and C. M. Lieber, One-dimensionalhole gas in germanium/silicon nanowire heterostructures, Proceedings ofthe National Academy of Sciences of the United States of America 102(29),10046–10051 (2005).

[4] U. Fano, Effects of Configuration Interaction on Intensities and Phase Shifts,Phys. Rev. 124(6), 1866–1878 (Dec 1961).




[8] H.-Y. Chiu, P. Hung, H. W. C. Postma, and M. Bockrath, Atomic-ScaleMass Sensing Using Carbon Nanotube Resonators, Nano Letters 8(12),4342–4346 (2008).

[9] B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and A. Bachtold, Ultrasen-sitive Mass Sensing with a Nanotube Electromechanical Resonator, NanoLetters 8(11), 3735–3738 (2008).

[10] Y. A. Pashkin, T. F. Li, J. P. Pekola, O. Astafiev, D. A. Knyazev, F. Hoehne,H. Im, Y. Nakamura, and J. S. Tsai, Detection of mechanical resonance of asingle-electron transistor by direct current, Applied Physics Letters 96(26),263513 (2010).

[11] B. Lassagne, Y. Tarakanov, J. Kinaret, D. Garcia-Sanchez, and A. Bachtold,Coupling Mechanics to Charge Transport in Carbon Nanotube MechanicalResonators, Science 325(5944), 1107–1110 (2009).

211

Appendix C

The effect of mechanicalresonance on Josephsondynamics

C. Padurariu, C.J.H. Keijzers, and Yu.V. Nazarov

We study theoretically dynamics in a Josephson junction coupled to a mechanicalresonator looking at the signatures of the resonance in d.c. electrical response ofthe junction. Such a system can be realized experimentally as a suspended ultra-clean carbon nanotube brought in contact with two superconducting leads. Anearby gate electrode can be used to tune the junction parameters and to excitemechanical motion. We augment theoretical estimations with the values of setupparameters measured in the samples fabricated.

We show that charging effects in the junction give rise to a mechanical forcethat depends on the superconducting phase difference. The force can excitethe resonant mode provided the superconducting current in the junction hasoscillating components with a frequency matching the resonant frequency of themechanical resonator. We develop a model that encompasses the coupling ofelectrical and mechanical dynamics. We compute the mechanical response (theeffect of mechanical motion) in the regime of phase bias and d.c. voltage bias.We thoroughly investigate the regime of combined a.c. and d.c. bias whereShapiro steps are developed and reveal several distinct regimes characteristic forthis effect. Our results can be immediately applied in the context of experimentaldetection of the mechanical motion in realistic superconducting nano-mechanicaldevices.

This chapter has been submitted for publication. See also http://arxiv.org/abs/1112.5807

C.I. Introduction

BM11902

REVIEW

COPY

NOT FOR D

ISTRIB

UTION

The Effect of Mechanical Resonance on Josephson dynamics

C. Padurariu, C. J. H. Keijzers, and Yu. V. NazarovKavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, The Netherlands.

(Dated: September 4, 2012)

We study theoretically dynamics in a Josephson junction coupled to a mechanical resonatorlooking at the signatures of the resonance in d.c. electrical response of the junction. Such a systemcan be realized experimentally as a suspended ultra-clean carbon nanotube brought in contact withtwo superconducting leads. A nearby gate electrode can be used to tune the junction parametersand to excite mechanical motion. We augment theoretical estimations with the values of setupparameters measured in one of the samples fabricated.

We show that charging effects in the junction give rise to a mechanical force that depends onthe superconducting phase difference and can excite the resonant mode. We develop a model thatencompasses the coupling of electrical and mechanical dynamics. We compute the mechanical re-sponse (the effect of mechanical motion) in the regime of phase bias and d.c. voltage bias. Wethoroughly investigate the regime of combined a.c. and d.c. bias where Shapiro steps are developedand reveal several distinct regimes characteristic for this effect. Our results can be immediately ap-plied in the context of experimental detection of the mechanical motion in realistic superconductingnano-mechanical devices.

PACS numbers: 85.85.+j, 74.45.+c, 73.23.Hk

I. INTRODUCTION

Nanoscale electromechanical systems (NEMS) convertsmall amplitude mechanical motion into measurable elec-trical currents1. Devices based on NEMS have found ap-plications as sensitive detectors of mass2, force3 and elec-trical charge4. Considerable research efforts have beendedicated to improving detection sensitivity by fabri-cating devices with higher resonance frequencies, lowerdamping rates (high quality factors) and larger couplingbetween electrical and mechanical degrees of freedom.

The problem of detecting the quantum state of amacroscopic mechanical resonator gave rise to severalmeasuring schemes, proposed5,6 as well as realized7. Im-provements in device fabrication have pushed the sen-sitivity threshold to the quantum limit7. In addition,new techniques for cooling mechanical motion have beenproposed8,9. The use of superconducting devices, in par-ticular, superconducting qubits to detect and control themechanical motion is in focus of modern research7,10. Itgives rise to a growing interest in techniques of couplingNEMS to superconducting circuits.

Superconducting nano-devices frequently use Coulombblockade that makes their transport properties sensitiveto the gate voltages. The same gate voltage can be usedto excite the mechanical motion which is detected fromthe change of d.c. transport properties of the device13–15.Without superconductors, this scheme has been success-fully realized for a metallic single-electron transistor17

and for a Coulomb-blockaded quantum dot in an ultra-clean carbon nanotube (CNT)15. The results revealedhigh resonance frequencies, reaching gigahertzs, and un-precedented quality factors of the order of 105. These de-vices can be made superconducting by connecting themto superconducting leads and providing sufficiently largecoupling between the states of the lead and device. We

have successfully realized Josephson junctions based onultra-clean CNTs. The supercurrent observed demon-strates a pronounced gate-voltage sensitivity that indi-cates a well-developed Coulomb blockade27.A very interesting proposal that combines Josephson

dynamics and mechanical resonator has been recently putforward by Gothenburg collaboration21. The authorsconsider an ideal ballistic CNT between two supercon-ducting leads biased at voltage V . Owing to Josephsonrelation, the current in the nanotube oscillates at fre-quency ωj = 2eV/�. The authors notice that in externalmagnetic field this gives rise to an oscillating Lorentzforce. If the frequency matches the frequency of the me-chanical resonator, the force excites mechanical motionwhich rectifies the Josephson current enabling the ob-servation of the effect in d.c. electric response of thejunction. One would observe a narrow current peak inI−V characteristics of the device. The same mechanismis responsible for Fiske steps25: the difference is that inFiske experiments the resonance is electrical rather thanmechanical.This provides us a motivation for the present theoret-

ical study. We address superconducting NEMS where amechanical resonator is integrated with a superconduct-ing circuit element, a Josephson junction. The details ofthe setup are given in Section II. In this work, we i. ex-plore the coupled dynamics of the oscillator displacementand superconducting phase difference and ii. describe themanifestations of mechanical motion in superconductingcurrent under various bias conditions. The goal is tolist experimentally observable effects. We investigate inparticular detail the effect of mechanical motion in thecontext of Shapiro steps as the most promising one withrespect to experimental detection.Part of our results are valid for any device combining

Josephson effect with mechanical resonance. However,we mostly concentrate on a class of particularly success-

215

C. The effect of mechanical resonance on Josephson dynamics

2

ful NEMS devices: suspended metallic CNT connected tosuperconducting leads. We have fabricated and studiedsuch devices. Since our research is theoretical we strivefor generality presenting thereby analytical and numeri-cal estimations and demonstrating reasonable values andranges of the parameters in use. Unfortunately, the merechoice of parameters to be in the reasonable range doesnot immediately quantify the values of the results ob-tained: they may vary significantly, and there is no sim-ple scaling present. The same is true for the experiments:the devices produced vary in length, in stress as indi-cated by measured frequencies, and in the conductancethat estimates the Josephson coupling. In this situation,we supplement general estimations with calculations fora set of concrete numerical values of the parameters inuse. This set is given in Subsection III F. This parameterchoice is therefore rather arbitrary, although it matchesclosely one of the devices made.

Studies of CNT Josephson junctions have shown thattheir Josephson energy can be modulated by the gate-induced charge on the CNT. Under these circumstances,the electrostatic energy of the system requires a specialconsideration (Subsection IIIA). With this, we formu-late the notion of Josephson mechanical force (SubsectionIII B). This is one of the main results of our work sincewe reveal a mechanism of phase-dependent mechanicaldriving which is different and generally more importantthan that considered in21. General equations of motionincluding this force (Subsection IIID) need to be fur-ther analysed to reveal which of the three competingnon-linearities is the most relevant one. The result ofrather involved analysis performed in Subsection IIIDshows that the most important non-linearity is the in-trinsic mechanical non-linearity, at least for CNT-basedsuperconducting mechanical resonators.

In fact, the mechanical non-linearity has been ignoredin21. In their case, the resonant enhancement of oscil-lating amplitude can only be stabilized by the feedbackfrom non-linear Josephson dynamics. This brought theauthors of21 to the analysis of strong feedback betweenthe displacement and superconducting phase. Our re-sults thus show that such feedback, although interesting,can not be realized in practical CNT devices since it isthe mechanical non-linearity that stabilizes the resonantgrowth of the oscillation amplitude.

Neglecting phase and charge non-linearities in com-parison with the mechanical one permitted us to sim-plify the equations drastically. Final equations and theworkflow to determine the quantity of interest — the me-chanical response current Im — are given in SubsectionIII E. Throughout the Section III we provide detailed es-timations of the displacement, force and electrical currentscales involved. It is our conclusion that the mechanicalresponse in our devices should be small. For the param-eter choice made, the mechanical response is at the scaleof 10−3 of the critical current.22

Further on, we apply the expressions obtained for avariety of bias conditions.

In Section IV we discuss the phase bias. We predictthe phase-dependent shift of the mechanical resonancefrequency that is an important signature of coupled dy-namics. Our estimation of the frequency shift shows thatit can be easily observable, being significantly larger thanthe resonance width. In addition, we show that a d.c.mechanical response current develops upon mechanicalexcitation of the device. We elaborate on the line shapeof the mechanical response current showing that for weakdriving conditions it is a usual Lorentzian dependence,while, in contrast, for strong driving conditions that in-duce a non-linear mechanical response, the line shapebecomes asymmetric, acquiring a Fano-type shape (Sub-section IVA).

Section V is devoted to d.c. voltage bias conditions.Resonant mechanical driving occurs at Josephson fre-quency matching the resonance frequency of the mechan-ical resonator, or an integer fraction of this frequency byhigher harmonics (Subsection VA). We study the re-sulting Fiske-type mechanical response and give the esti-mations of the effect. For completeness, we also shortlydiscuss the possibility of parametric excitation. We showby estimation that the emergence of parametric mechani-cal response requires a large Josephson energy, at least anorder of magnitude larger than that achievable in practi-cal devices (Subsection VA).

Sections VI and VII are devoted to the dynamics in thepresence of external a.c. drive, in the regime where theJosephson junction gives rise to well-developed Shapirosteps26. One of the motivations of the use of Shapirosteps is the better synchronization conditions in compar-ison with d.c. voltage bias. This can be seen as fol-lows. The big quality factor Q of the nanomechanicalresonance results in a narrow Fiske-type current peak(discussed in section V). Its width in voltage can be es-timated as δV � V/Q. The observation of such a narrowfeature imposes a severe limitation on voltage noise SV :to resolve the peak one must achieve SV � (e/�)V/Q.This may be challenging under realistic experimental cir-cumstances. There is a way out: the voltage can besynchronized with the frequency of external irradiation.This happens at Shapiro steps and effectively reduces thevoltage noise.

In Section VI we study the mechanical response at theShapiro steps in the regime where the a.c. driving fre-quency matches the resonant frequency, and present thepeculiarities of this response. We derive explicit analyti-cal expressions of the mechanical response as a functionof a.c. driving amplitude and illustrate them with plots.We concentrate on the effect seen in d.c. current mea-surement, that is, the modification of the width and posi-tion of Shapiro steps. We show that the response at thefirst Shapiro step (Subsection VI A), which develops atthe position of the Fiske-type current peak, is qualita-tively different from that at the higher Shapiro steps (VIB). In Section VII we present the same considerations forthe case of the non-resonant driving that in the regime ofShapiro steps can efficiently excite the mechanical motion

216

C.II. The setup

3

owing to Josephson non-linearities.Our preliminary experimental results show correspond-

ing features. They will be presented elsewhere27 uponcompletion of detailed analysis and comparison with ourtheoretical findings. Our concluding remarks are pre-sented in Section VIII.

II. THE SETUP

The setup under consideration is sketched in Figure 1where we concentrate on a case where both the Josephsonjunction and the mechanical resonator are realized withthe same single CNT. Even in this case, the couplingbetween mechanical and electrical degrees of freedom isrelatively weak. This allows us to describe the electri-cal and mechanical aspects of the setup separately. Weprovide the description in this Section, while in the nextSection we concentrate on the coupling.

A. Electrical Setup

We consider a conducting link between two supercon-ducting leads (the CNT in Fig. 1). In general, the cur-rent flowing in this junction is a complicated non-linearand time-delayed function of superconducting phase dif-ference between the leads. However, we assume that inthe relevant frequency range the current response of thejunction is superconducting and instant.The junction is included into an external electric cir-

cuit and the voltage drop at the junction is related to thetime derivative of Josephson phase, ϕ = 2eV/�. In gen-eral, a circuit that connects the leads can be describedby a complex frequency-dependent impedance Ze(ω) inseries with a voltage source Vb. We typically assumethat Ze by far exceeds the typical junction impedanceat low frequencies while at frequency scale of Joseph-son generation frequency ω � eV/�, Ze(ω) is negligi-ble in comparison with the junction response. In thiscase, the junction is current-biased at low frequencieswith Ib = Vb/Ze(0) and voltage-biased at Josephson fre-quencies. While this scheme looks different from the tra-ditional resistively shunted junction model (RSJ) wherethe external impedance is connected in parallel and thejunction is current-biased, it is equivalent to a generalizedRSJ upon transforming the impedance and the voltagesource. For instance, the linear part of possible quasipar-ticle response of the junction can be incorporated intoZe(ω).In addition, the junction is affected by the gate elec-

trode biased by voltage source Vg . The bias and gatecircuits are disconnected at zero frequency. At finite fre-quency, there is a cross-talk between the circuits which isdifficult to eliminate or even characterize in realistic ex-perimental circumstances. We account for that by corre-lating a.c. parts of the voltage sources Vb,g. For instance,if the gate voltage consists of a d.c. part and a harmonic

signal at frequency Ω, Vg(t) = Vg0 + Vg cos(Ωt + χ), thebias voltage source should also oscillate at the same fre-quency, Vb(t) = Vb0 + Vb cos(Ωt). The ratio of two a.c.amplitudes and their mutual phase shift χ is determinedby details of the crosstalk. We will show below that theinterference of these two a.c. signals may strongly affectthe d.c. currents in the junction.The superconducting current is determined by the in-

stant phase difference, I(t) = I(ϕ(t)). In this case, it isrelated to the Josephson energy Ej of the junction,

I = (2e/�)∂Ej(ϕ)/∂ϕ . (1)

It is essential for us that the Josephson energy is notonly a function of phase difference but also depends onthe gate voltage through the charge q = CgVg induced inthe resonator, Ej = Ej(q, ϕ).For a nanotube device, the origin of this charge sen-

sitivity is (weak) Coulomb blockade of electrons in themiddle of the nanotube. The nanotube can be viewedas two junctions in series, those being formed at contactwith metallic leads. If the conductance of the junctionsis smaller or comparable with the conductance quantumGQ ≡ e2/(π�) ≈ 7.75× 10−5 Ω−1 Coulomb interactionsbecome important and set a quasiperiodic dependence ofJosephson energy on q with a period 2e. This correspondsto charge quantization in the middle of the nanotube. Weroutinely observe the quasi-periodic modulation of super-conducting currents in fabricated nanotube devices. Themodulation can be tuned by changing the gate voltageat scale of q � 10 − 100e from values of the order of 1to several per cent. Big modulation and well-developedCoulomb blockade require big junction resistances, thisstrongly suppresses the superconducting current. It istherefore advantageous to have intermediate resistancesR � G−1

Q . At R = 5 kΩ we typically observe 30% mod-ulation.The superconducting current is a periodic function of

the phase I(ϕ) = I(ϕ + 2π) and therefore can be ex-panded in harmonics as23:

I(q, ϕ) = I1(q) sin(ϕ) +

∞∑n=2

In(q) sin(nϕ) , (2)

If one neglects all harmonics except the first one, I1gives the critical current of the Josephson junction. Wewill typically assume this, and will mention the effect ofhigher harmonics only if it is crucial.

B. Mechanical Setup

Mechanical resonators can be realized in a variety ofways11. In many cases the adequate description of theresonator can be achieved with a minimum model thataccounts for excitations of a single resonator mode, ne-glecting coupling to any other modes. The minimummodel is given by the following equation of motion for a

217


4

FIG. 1: (Color online.) The setup. The sketch presents themechanical resonator realized as a CNT suspended over twosuperconducting leads isolated from the back gate electrode.The CNT center is displaced in y-direction by an electrostaticforce produced by the gate voltage. The superconductingleads are parts of an electrical circuit characterized by animpedance Ze. The setup can be biased by either voltage orcurrent source.

displacement variable y:

y + Γy + ω20y − αy2 − βy3 = F (t)/M . (3)

Here F (t) is the time-dependent driving force, M is theeffective mass corresponding to the mode, ω0 stands forthe resonant frequency, Γ � ω0 is the damping rate,and β is the parameter describing the leading cubic non-linearity12. The cubic non-linearity provides the impor-tant restriction on the magnitude of the displacement atresonant frequency as a reaction on resonant force. Wealso keep the second-order non-linearity α. Although itis not important in analysis of the reaction at resonantforce, it describes the shift of the resonant frequency dueto constant force.Our preferable realization of mechanical resonator is a

suspended ultra-clean CNT13–15 that demonstrates bestquality factors observed so far(Q ≡ ω0/Γ � 105). Inthis Subsection we review the parameters of the minimalmodel for this realization. In the setup shown in Figure1, the nanotube displacement from equilibrium positionand the driving force are in the y-direction towards thegate, that is, perpendicular to the nanotube axis. Themechanical variable y(t) is the displacement of the mid-point of the nanotube.In the case of a CNT, the adequate model of me-

chanical properties involves a suspended thin cylindricalrod clamped at both ends where the nanotube touchesthe metal leads. The parameters are the rod length L,the cylinder radius r, and the tube cross section areaS. In our experiments, L � 0.3..0.5 μm, r � 1..2 nm,and S � 2πra � 2.1..4.3 nm2 for a single-wall nan-otube, a � 0.34 nm being the layer spacing in graphite.The relevant elasto-mechanical material constants, car-bon Young’s modulus E and graphite density ρ are es-timated as E � 1012 J/m316 and ρ � 2.2 g/cm3.The bending modes of the rod and their complete dy-

namics are described by the Euler-Bernoulli equation ofmotion19,20.We concentrate on the lowest frequency bending mode

that has no nodes in the rod and therefore is easy to ex-cite. The resonant frequency can be tuned by ”tighten-ing” the tube, that is, changing the elastic tension. Thisis achieved by applying a sufficiently big d.c. gate voltageVg0. The resulting electrostatic force strives to elongatethe nanotube, thus producing the tension. In such a way,the resonant frequency can be increased by a factor ofthree in comparison with that of the ”loose” nanotube.For estimations, we concentrate on the case of loose rod.In this case, the resonance frequency corresponding to thelowest CNT bending mode can be estimated in terms ofthe bending spring constant and the carbon mass densityω0 � 22.4

√EI/ρSL−2, where I � Sr2/2 is the moment

of inertia18 of the CNT cross section I =∫x2dS, as in-

troduced in19. In our devices of length L = 0.3..0.5 μm,the frequency is ω0/2π � 0.30..0.84 GHz, similar to fre-quencies reported in15. The effective force is evaluatedusing the eigenfunction of the mode ξ(x) ≡ y(x, t)/y(t),

F =∫ L

0 dxf(x)ξ(x), f(x) being the force per unit length.For electrostatic forces, an ad-hoc assumption is that theforce distribution is uniform, so the total force is Ft = fL.In this case, F � 0.53Ft. The effective mass is given by

M = ρS∫ L

0 dxξ2(x), M � 0.41ρSL � 4.1..6.8 × 10−22

kg. The cubic non-linearity originates from the tensionproduced by the nanotube displacement, the correspond-ing parameter can be estimated as β � 40 ES/ML3 �ω20/r

2 � 1.8..5.5 GHz2nm−2, assuming uniform distribu-tion of force along the length of the rod. The second-order non-linearity α vanishes for loose straight rod forsymmetry reasons. However, it becomes significant if therod is tightened such that its frequency change with re-spect to the loose rod value ω0 is of the order of ω0. Inthis case, the non-linearity is obtained as α = 3βy0, y0being the equilibrium displacement induced by the tight-ening.If F (t) oscillates at frequency ω close to the resonant

frequency, Eq. (3) can be solved in resonant approxima-tion for the complex amplitude y:

y =F

2Mω0

−1

ν + iΓ/2 + (β′/2ω0)|y|2 , (4)

with F being the complex force amplitude. Here weintroduce the detuning ν ≡ ω − ω0 implying that|ν| � ω0. We also introduce the Duffing parameterβ′ = β+α2/ω2

0 � ω20/r

2 corresponding to the amplitude-dependent frequency shift. In our experiments, we esti-mate β′ � 3.6..11 GHz2nm−2.We will re-write Eq. (4) in dimensionless form intro-

ducing a critical amplitude yc, yc =√ω0Γ/β′. At this

amplitude scale, the response of the resonator becomes atwo-valued function of detuning (see Fig. 2) For a CNT,it can be estimated as y2c � r2/Q which correspondsfor our experiments to yc � 3.2..6.4 pm. The drivingforce corresponding to yc is Fc = Mβ′y3c = Mω2

0yc/Q.

218

C.III. Coupling and non-linearities

5

FIG. 2: (Color online.) Left panel: Real part of the complexdisplacement amplitude Re y versus detuning. Right panel:Imaginary part of the complex displacement amplitude Im yversus detuning. The three curves in each panel correspondto |F |/Fc = 0.5, 1, 2.

We estimate it for a CNT Fc � 102 ES(r/L)3Q−3/2 �(1.2..2.4)× 10−18 N.The dimensionless form of Eq. (4) is:

y

yc=

F

FcR

(ν

Γ,F

Fc

); R(a, b) =

−1

2a+ |b|2|R(a, b)|2 + i

(5)

Here we have introduced a dimensionless complex re-sponse function R(ν, F ). In the linear regime |F | � Fc

its dependence on F can be neglected: R(a, b) = (2a +i)−1.Figure 2 shows the real and imaginary parts of y as a

function of detuning for three values of the driving forceamplitude that correspond to quasi-linear, critical andbistable regimes.

III. COUPLING AND NON-LINEARITIES

In this Section, we analyze the coupling between me-chanical and electrical degrees of freedom. The couplingmanifests as two quantities: displacement-dependent cur-rent and phase-dependent mechanical force. Both quan-tities emerge from electrostatic effects, therefore we willstart with a detailed discussion of electrostatic energy inthe setup, finding the induced charge for a given mechan-ical displacement and phase. We then use the inducedcharge to express the forces and superconducting current.By doing so, we assume that the typical time of chargeequilibration is much shorter than the typical time scaleω−10 of the mechanical motion. We compare the electro-

static phase-dependent force with Lorentz force proposedin21. We derive the coupled equations of motion govern-ing the Josephson and mechanical dynamics and identifythe dominant source of non-linear behavior.

A. Electrostatic energy

The junctions connecting the middle of the nanotubeto the leads have intermediate resistance, so that themiddle of the nanotube forms a Coulomb island that

is neither isolated from, nor ideally connected to theleads. While the case of good isolation20 can be eas-ily treated microscopically, the situation of intermediateconductances is difficult to quantify from a microscopiccalculation. However, the situation can be completely an-alyzed at the phenomenological level. At this level, theanalysis is a case of elementary non-linear electrostatics.In comparison with20, the analysis adds some importantand less obvious details, so we choose to outline it at acomprehensive level.To start with, let us assume that the capacitance to the

gate is vanishingly small while Vg is diverging such thatthe charge induced to the Coulomb island by the gateq = CgVg tends to a constant limit. A part of the ground-state energy of the setup, Ec(q), depends on q. GeneralCoulomb-blockade considerations24 imply that this partis a (quasi) periodic function of q with a period of 2e. Inthe limit of full isolation, for instance, this energy is piece-wise parabolic, Ec(q) = ECminN (N − q/2e)

2, N being

an integer number of extra Cooper pairs stored in theisland. In general, it is a smooth function of q and maydepend on the superconducting phase difference ϕ and,in principle, on mechanical displacement y. This energyresults in a non-zero electrostatic potential difference be-tween the island and the leads, V (q) = ∂Ec(q)/∂q.Let us now turn to finite Cg and therefore finite Vg

that is the potential difference between the leads and thegate electrode. Since this is not the potential differencebetween the island and the gate anymore, the inducedcharge q is not equal to CgVg. Rather, it is determinedfrom the voltage division between two capacitors: Cg andone between the island and the leads. The total voltagedifference Vg is the sum of the voltage drops at the twocapacitors,

Vg =q

Cg+ V (q).

The charge is then found from this equation that can berewritten as

q = CgVg − Cg∂Ec(q)

∂q. (6)

We note that this is equivalent to the minimization ofthe total electrostatic energy with respect to q,

E = minq

(Ec(q) +

q2

2Cg− qVg

). (7)

Indeed, the condition of the energy minimum coincideswith Eq. (6).There are two implicit dependences in this equation

that distinguish it from pure electrostatics, and that wemake explicit now. First of all, the electrostatic energydepends on the mechanical displacement of the nanotube.Geometric considerations suggest that this dependencecan be ascribed to Cg: indeed, the modification of capac-itance to the gate is linear in y, Cg → Cg + (dCg/dy)ywhile the modification of Ec is expected to be ∝ y2.

219


6

Secondly, the electrostatic energy depends on the super-conducting phase difference: indeed, the Josephson en-ergy is just the phase-dependent part of Ec, Ec(q, ϕ) =Ec(q)+Ej(q, ϕ). The electrostatic charge q depends bothon displacement y and on superconducting phase ϕ.We note at this point that we can skip Ec(q) from the

total energy and replace Ec(q, ϕ) with Ej(q, ϕ) even ifEj(q, ϕ) � Ec(q). The reason is that the phase indepen-dent term Ec(q) results only in a small offset of q fromits value of CgVg. As far as the offset does not depend onphase, we can disregard it. The phase-dependent part,however, is important — it provides the coupling betweenthe mechanical and Josephson subsystems.To single out these contributions, we assume that i.

the voltage between the middle of the nanotube and theleads is smaller than the gate voltage, ∂Ec/∂q � Vg,this is fulfilled if the induced charge q � e, i.e., inany practical setup; ii. the mechanical displacementis small in comparison with the distance to the gate,

y � Cg

(dCg

dy

)−1

� Lg (in our experiments Lg � L � y).

With this, we linearize Eq. (6) with respect to theJosephson energy and mechanical displacement to arriveat (q0 ≡ CgVg)

q = q0 + VgdCg

dyy − Cg

∂Ej

∂q(q, ϕ) (8)

The first term is the common expression for the gate-induced charge while the second and the third are thecorrections of interest. At the moment , we keep q inthe argument of Ej , although q ≈ q0. The point is thatthe Josephson energy is sensitive to variations of q ofthe order of e, and (q − q0) can in principle be of thisorder. Since y � Lg, we may disregard the possible ydependence of dCg/dy.

B. Coupling quantities

It is thus the charge-dependence of current and forcethat gives rise to the coupling between the supercon-ducting and mechanical dynamics. This dependence isin general complicated containing both linear and non-linear terms. The importance of non-linear terms is de-termined by comparing the resulting non-linear feedbackwith intrinsic non-linear terms characterizing the Joseph-son and mechanical dynamics and will be addressed inthe next Subsection. In this Subsection, we anticipatethat the charge and phase non-linearities are unimpor-tant and give the coupling terms linearising the charge-dependence. We need to discuss i. the displacement-dependence of current and ii. the phase-dependence offorce.i. It is convenient to separate the superconducting cur-

rent I(q) into the static component I(q0) and the compo-nent which is linear in charge variations (dI/dq)(q− q0).We are interested in a (d.c.) current response on the me-chanical motion, the mechanical response. It arises due

to the direct modulation of charge by the mechanical dis-placement,

Im(t) =∂I

∂q

dCg

dyVg0 y(t) (9)

where we used Eq. (8) to express the displacement-dependence of charge.We will mostly concentrate on the situation when the

displacement oscillates at the resonant frequency while ad.c. component of Im is of interest. The d.c. signal thenarises from the rectification of y(t) by an oscillating partof ∂I(ϕ)/∂(q/e), that we call the detecting current.ii. The mechanical resonator is affected by the electro-

static force F = −∂E/∂y:

F =dCg

dy

q2

2C2g

, (10)

where we have used the expression of E given in Eq. (7).It is convenient to distinguish three separate contribu-

tions to the total force: the static force, the gate drivingforce and the phase-dependent Josephson force.The static force is produced by the d.c. gate volt-

age. Its magnitude is given by Fst = (dCg/dy)V2g0/2,

corresponding to the first dominating term in Eq. (8).The effect of the static force is to tighten the resonator,thereby tuning its resonance frequency15. Since it is sta-tionary it does not excite the oscillations.The a.c. gate driving force arises due to the a.c.

modulation of the gate voltage and is given by Fg =

(dCg/dy)Vg0Vg � (q0/e)(eVg/Lg).The phase-dependent Josephson force, not discussed in

previous literature, comes about the product of the firstand third term in Eq. (8),

Fj = − dCg

dyVg0

∂Ej(q, ϕ)

∂q, (11)

In fact, it is similar to the gate driving force, with Vg

replaced by the phase-dependent voltage arising in thecapacitive network, ∂Ej(q, ϕ)/∂q. In contrast to the gatedriving force, the time dependence of the Josephson forceis determined by the phase dynamics rather than theexternal modulation of the gate voltage.The scale of the Josephson force is Fj � (q/e)(Ej/Lg),

where Ej is the charge-dependent part of the Joseph-son energy, which for intermediate contact conductances� GQ represents a fraction of � 10..50% of the to-tal Josephson energy. The Josephson force scale canbe compared to the scale of a.c. gate driving force,Fj/Fg � Ej/eVg. For sufficiently low a.c. driving ampli-

tude eVg � Ej the Josephson force dominates Fg � Fj .

C. Lorentz force

The Josephson force explained above arises from thecombined effect of charge sensitivity of the Josephson

220

C.III. Coupling and non-linearities

7

coupling and the capacitive coupling to the gate elec-trode. The alternative mechanism of generating a ϕ-dependent force was recently proposed by G. Sonne etal21. This force is of Lorentz type arising from the inter-action of the ϕ-dependent superconducting current with

an external magnetic field �B applied in the perpendicu-lar direction. (for the setup of Figure 1, along the z-axis)This mechanism does not require the presence of a gate.Let us compare the Lorentz force and the electrostatic

Josephson force. The Lorentz force is in y-direction, that

is, perpendicular to both �B and the superconducting cur-

rent, FB = L| �B|I. It is natural to express Fj in terms of

the electric field | �E| = (dCg/dy)Vg/Cg produced by thegate electrode. For estimates, we assume L � Lg and(dCg/dy)L/Cg � L/Lg � 1. This yields

FB

Fj� c| �B|

| �E| α , (12)

where c � 3 × 108 m/s is the speed of light and α =e2/4πε0�c � 1/137 is the fine structure constant.

Typical magnetic fields used in experiments are | �B| � 1T . They are limited from above by the critical fields ofthe superconducting leads. The typical electric fields are

| �E| � 107 V/m. This corresponds to a potential drop ofVg � 10 V over a distance of Lg � 0.5 μm. For thesevalues FB/Fj � 10α � 1 suggesting that the Josephsonforce dominates. Therefore in the rest of the paper wewill disregard the Lorentz force.If one imagines a ballistic nanotube, the Josephson

coupling is not affected by the induced charge. In thiscase the Lorentz force would be the only superconduct-ing phase-dependent driving mechanism. However, theideally ballistic nanotubes have not been realized exper-imentally.

D. Analysis of non-linearities

Let us bring together three coupled equations govern-ing the dynamics of the setup

Vb(ω)

Ze(ω)+ i

�ω

2e

ϕ(ω)

Ze(ω)= (I(q, ϕ))ω , (13)

y + Γy + ω20y + αy2 − βy3 = M−1 dCg

dy

q2

2C2g

, (14)

q0 +q0Cg

dCg

dyy − Cg

∂Ej

∂q(q, ϕ) =q . (15)

The first equation describes the dynamics of supercon-ducting phase difference ϕ(t) and is obtained by applyingKirchhoff’s laws to the circuit. The second equation is forthe mechanical displacement y(t) where we substitute theelectrostatic force given in Eq. (10). The induced chargeq enters both equations, and at the same time is definedby the third equation, that is, its value depends both onϕ and y. Therefore the equations are coupled.

The equations of motion can be derived using a La-grangian or Hamiltonian method. This we present inAppendix A.We wish to simplify these equations under experimen-

tally relevant assumptions. For this, we shall analyzethe relative importance of different non-linearities in thecoupling. There are natural non-linearity scales for allthree variables, ϕ � 2π, q � e, y � yc � √

ω0Γ/β′.This could change if the coupling is sufficiently strong.For instance, the displacement may cause the variationof phase that is subject to Josephson non-linearity. Theresulting variation of phase would produce the non-linearvariation of q. This will result in non-linear feedback ony and could in principle give rise to a non-linear scaleof y that would be smaller than yc. Therefore, first ofall we shall quantify the coupling between electrical andmechanical variables by comparing the non-linear termsin the mechanical force resulting from the coupling withthose coming from the intrinsic non-linearities character-ized by α and β.The conclusion of this Subsection is that the mechani-

cal non-linearity is dominant. We prove this with a ratherinvolved reasoning given below.For the estimations, it is convenient to introduce the

following dimensionless parameters:

Aj = Cg∂2Ej

∂q2� Ej

EC, EC ≡ e2

Cg

Bj =2e2Ze

�ω0

∂I

∂q� GQZe

Ej

�ω0

For estimations, we assume that Aj , Bj are either smallor of the order of 1. This assumption is valid for Aj ;it compares the Josephson energy to the charging en-ergy under conditions of well-developed Coulomb block-ade. The parameter Bj is a coefficient of Josephson feed-back at high frequency and depends on the details of theexternal circuit via the impedance Ze. Unless a specialeffort is made to increase the circuit impedance at highfrequency, Bj will not be big.Let us estimate the linear responses of the charge, δq,

and the superconducting phase δϕ on a given displace-ment variation δy. We do this by expanding Eqs. (15)and (13) up to linear terms in δq, δϕ, δy and expressingδq, δϕ in terms of δy. This yields

δq (1 +Aj +AjBj) = q01

Cg

dCg

dyδy � q0

δy

Lg(16)

δϕ (1 +Aj +AjBj) = Bjq0e

1

Cg

dCg

dyδy � Bj

q0e

δy

Lg(17)

Since in addition we assume that Aj , Bj � 1, these linearresponses can be estimated as simple as

δq � q0δy

Lg, δϕ � Bj

q0e

δy

Lg.

We use this to find a scale y1 at which the responsesof charge and superconducting phase may become com-parable with the scales of their intrinsic non-linearities

221


8

δq � e and δϕ � 2π. To do this, we substitute δq � e,δϕ � 2π, δy � y1. Both equations lead to the sameestimation y1 � Lg(e/q0).We need to compare this scale with the scale yc �

r/√Q of the mechanical non-linearity. This yields

y1yc

� e

q0

Lg

r

√Q (18)

Two last factors in this expression are big, while the firstone can be small. We estimate the biggest q0 from thecondition that ω0 is changed significantly by applying thegate voltage, that is, the resonator is tightened, whichleads to the estimation of the stationary displacementy0 � r. This yields

q0e

� r2L1/2a3/2, (19)

a being atomic scale, q0/e � 100 for our devices. Withthis, we estimate the first two factors as (e/q0)(L/r) �(aL/r2)2. This is � 10 for our geometries and we con-clude that y1/yc � 1 for any Q > 1. This implies thatupon increasing the magnitude of the oscillations δy weencounter the mechanical non-linearity first, and can dis-regard other non-linearities at this magnitude scale.This proves that the dynamics of charge and phase is

linear in y provided our estimations of mechanical non-linearities α, β hold.We still need to show that the coupling to Joseph-

son junction does not change these non-linearities sig-nificantly. To this end, we estimate the quadratic andcubic non-linearities of the mechanical force due to cou-pling. First we find the quadratic and cubic variations ofthe charge with respect to displacement using Eq. (15).

δq(2) = Cg∂3Ej

∂q3(δq)2 � eAj

(q0e

δy

Lg

)2

, (20)

δq(3) = Cg∂4Ej

∂q4(δq)3 � eAj

(q0e

δy

Lg

)3

. (21)

We can now estimate the terms in the mechanical forcethat are quadratic and cubic in δy.

δF (2) =dCg

dy

q202C2

g

(2

(δq

q0

)2

+δq(2)

q0

)

� Fst

(1 +Aj

q0e

)(δy

Lg

)2

, (22)

δF (3) =dCg

dy

q202C2

g

(3δq

q0

δq(2)

q0+

δq(3)

q0

)

� FstAj

(q0e

)2(δy

Lg

)3

. (23)

We compare these terms with the intrinsic non-linearities. The second order term δF (2) needs to becompared with the mechanical quadratic non-linearityMαδy2. Assuming the static displacement of the order

of CNT radius, y0 � r, we estimate Mα � Mβ/r �Mω2

0/r.

δF (2)

Mαδy2�

(1 +Aj

q0e

) r2

L2g

� 1 . (24)

Here we use the estimation (q0/e)(r/Lg) � 1,(q0/e)(r/Lg) � 0.1 for typical CNT geometries. (see Eq.

(19)). The third order term δF (3) needs be comparedwith the third-order non-linearity Mβδy3. This yields

δF (3)

Mβδy3� Aj

r3

L3g

� 1 . (25)

To summarize, we proved that the non-linear scalescorrespond to ϕ � 2π, q � e, y � yc � √

ω0Γ/β′ andthat for a CNT resonator the intrinsic mechanical non-linearities dominate the non-linearities arising from cou-pling. This permits a simplification of the dynamicalequations. We may linearize the terms describing thecoupling of mechanical displacement and electricity, thusseparating Josephson and mechanical non-linearities.

E. Workflow

This sets the following workflow.

• At given a.c. and d.c. bias and gate voltages wesolve for Josephson dynamics neglecting the me-chanical coupling and setting q = q0(t). We findI(t) and ϕ(t). Using these, we compute the Joseph-son force Fj given by Eq. (11).

• We solve the non-linear mechanical equation

M(y + Γy + ω20y + αy2 − βy3) = Fst + Fg + Fj . (26)

to find y(t). We are mostly interested in a partthat oscillates with frequency � ω0. This may beexcited by both Fj and Fg.

• We calculate the mechanical response current de-fined in subsection III B,

Im(t) =2e

�

∂2Ej

∂ϕ∂q

dCg

dyVg0 y(t) . (27)

• If we can neglect the feedback in Josephson dy-namics, we are done, since the response is givendirectly by Im. In general, there is such a feed-back since change of the current results in a corre-sponding change of phase. The condition to neglectthe feedback is the condition of phase bias at fre-quency � ω0, that is, the inductive impedance ofthe junction � �ω0/(GQEj) is much bigger thanthe impedance Ze(ω0) of the external circuit. Thisis the case for devices fabricated so far.

To account for the feedback, we linearise theJosephson dynamics to determine the response of

222

C.IV. Phase bias

9

the superconducting phase on the mechanical re-sponse current found, ϕm(t)

ϕm(t) =�

2e

∫ t

dt′dt′′Z(t′, t′′)Im(t′′)

Here the kernel Z(t, t′) represents the combined lin-ear impedance of the external circuit and the junc-tion. It follows from the discussion in subsectionIIID that ϕm(t) � 2π.

• Taking this into account, we obtain the currentresponse sought:

Im = Im +2e

�

∂2Ej

∂ϕ2ϕm (28)

The first term is the direct modulation of the cur-rent by the charge induced by the mechanical dis-placement while the second one is the feedback ofthe Josephson junction by means of ϕm.

The mechanical response is thus typically a small cor-rection to the maximum superconducting current. Wecan estimate it at maximum taking the

ImIc

� q0e

ycL

� q0r√QeL

� 10−3

(We remind that (q0/e)(r/L) � 0.1, Q � 105 for ourdevices). Perhaps unexpectedly, the typical response be-comes smaller upon increasing the quality factor. Thereason for this is clear: the maximum displacement be-comes smaller. However, the large Q results in sharpfrequency dependence of the response making it easier toidentify. We thus concentrate on this dependence.

F. Parameters

Let us specify a concrete choice of the values of param-eters we will use for numerical estimations. We choosethese values to match those of one of the devices fabri-cated. Yet we shall stress that the choice made is ratherarbitrary and relatively small deviations in each param-eter can accumulate changing the estimations of the me-chanical effect by orders of magnitude.

Junction critical current, Ic : 1.0× 10−8 A; (29)

Total Josephson energy, Ej : 2.1× 10−5 eV;

Static gate voltage, Vg0 : 1.0 V;

Static charge on the resonator , q0/e : 100;

Resonator length and distance to gate, L = Lg : 0.3 μm;

Resonator mass, M : 4.1× 10−22 kg;

Resonance frequency, ω0/2π : 0.84 GHz;

Quality factor, Q : 1.4× 105;

Quadratic non-linearity, α : 5.5 GHz2nm−1;

Cubic non-linearity, β : 5.5 GHz2nm−2;

Scale of maximum displacement, yc : 6.3 pm;

Mechanical force scale, Fc : 2.3× 10−18 N;

Josephson force, Fj : 1.1× 10−15 N.

Two important dimensionless parameters are relativevalue of the mechanical response current, Im/Ic, andthe ratio of maximum Josephson force to the mechanicalforce scale, Fj/Fc. As mentioned, the response is rela-tively small, Im/Ic � 10−3. In contrast to this, Fj/Fc isbig, Fj/Fc � 500.

This means that the Josephson force can easily drawthe oscillator to very non-linear regime. Let us note thatthe charge-dependent part of Ej is only a fraction of thetotal Josephson energy, say, 10%. This gives more real-istic estimation Fj/Fc � 50 that we will use in the plots.Besides, the oscillatory dependence of Ej on q permitstuning Fj to zero.

IV. PHASE BIAS

Let us start our considerations with the junction bi-ased with a time-independent phase ϕ: such bias condi-tion can be achieved by embedding the junction into asuperconducting loop. Unfortunately, our present exper-imental setup does not allow measurements under thesebias conditions. We present the theoretical results inhope that they will be useful for future experiments.

The simplest experimental signature of Josephson forceunder phase bias conditions is the phase-dependent shiftof the resonant frequency. The mechanism of this shift inour situation is the mechanical non-linearity: the staticJosephson force tightens or looses the nano-tube resultingin the frequency change. The frequency response on thestatic force in our model reads

dω0

dF=

α

Mω30

� ω0

Fst. (30)

223


10

The phase-dependent frequency shift reads

Δω0(ϕ) =∂ω0

∂FFj(ϕ) = − α

Mω30

dCg

dyVg0

∂Ej(q, ϕ)

∂q

� ω0Fj(ϕ)

Fst� ω0

Ej

eVg0cos(ϕ) . (31)

The shift is clearly observable provided it exceeds thebroadening Γ. The estimation gives

Δω0(ϕ)

Γ� Q

Ej

eVg0cos(ϕ). (32)

For the parameter set chosen, Eq. (29), the maximumvalue of the frequency shift is(

Δω0(ϕ)

Γ

)max

= 2.8 . (33)

Therefore, the shift is clearly observable.Let us consider an example of a mechanically-induced

response under conditions of the phase bias. To excitemechanical oscillations, we apply an additional a.c. volt-age to the gate that oscillates at the frequency Ω close tothe resonant frequency ω0.

Vg = Vg0 + Vg cos(Ωt) . (34)

Assuming Vg to be sufficiently small to provide a linearresponse of the displacement, we obtain the following ex-pression for the resonant part of the displacement:

y =Fg

2Mω0

−1

ν(ϕ) + iΓ/2, Fg =

dCg

dyVg0Vg , (35)

y(t) =1

2

(ye−iΩt + y∗eiΩt

), (36)

Here, ν(ϕ) = Ω−ω0−Δω0(ϕ) � ω0 is the detuning thatincludes the phase dependent shift of the resonance fre-quency discussed above. Owing to the mechanical non-linearity, the oscillating displacement produces a station-ary displacement y = α|y|2/ω2

0. This induces a station-ary charge that affects the d.c. superconducting currentat constant phase bias. Rather remarkably, this effect isrelated to the phase-dependent shift discussed above. In-deed, both are proportional to charge-dependent part ofthe Josephson energy and to the non-linearity coefficientα. The resulting current response reads

Im = −2e∂

∂ϕ(Δω0(ϕ)) |y|2(Mω0/�), (37)

the contribution to the current being of the order of eΔω0

if the displacement magnitude is of the scale of quantumfluctuations

√�/Mω0. The dependence on frequency of

the a.c. modulation is a Lorentzian one, as it is frequentlyexpected (Fig. 3), the Lorentzian center being shiftedwith changing the phase.

FIG. 3: (Color online.) The phase-dependent frequency shiftfor the case of weak driving. The curves give the linear re-sponse of d.c. current in units Ia = 2e(Δω0)max|y|2(Mω0/�)as a function of frequency detuning for a set of phase biasvalues: from the lowermost to the uppermost curve the phasechanges from ϕ = π/8 to ϕ = 15π/8, with interval π/8. Thecurves are offset for clarity. Dashed lines give the positions ofzero.

A. Fano-type response

The above mechanism of response exploits the domi-nating mechanical non-linearity. It is proportional to y2.Upon increase of the a.c. amplitude Vg the oscillatingdisplacement saturates owing to the non-linearities. Inthis case, the dominating d.c. current signal can arise asa result of mixing of the oscillating displacement y andthe oscillating charge ∝ Vg. The resulting current is thus

proportional to Vg y and may exceed the contribution ∝ yprovided the latter saturates.The expression for this contribution reads

Im =2e

�

∂3Ej(q, ϕ)

∂q2∂ϕCg

(dCg

dyVg0Re

{Vg y

})(38)

Interestingly, it exemplifies a Fano-type dependence onthe detuning that is quite different from a Lorentzian. Inthe linear regime,

Im(ν) ∼ Γ

2

ν(ϕ)

ν(ϕ)2 + Γ2/4, (39)

so that the signal changes sign at the resonance point.Fig. 4 illustrates the Fano-type dependence in the non-linear regime. Comparing expressions (39) and (37) weconclude that the Fano-shaped Im dominates providedVg/Vg0 � (q0/e)

−3/2 � 10−3. The condition occurs deepin the non-linear mechanical response regime.

V. D.C. VOLTAGE BIAS

Let us turn to d.c. voltage bias. In this case, thesuperconducting phase is in first approximation a linear

224

C.V. D.C. voltage bias

11

FIG. 4: (Color online.) An example of Fano-type frequency-dependence of the mechanical response (38). The curves cor-

respond to the driving force values |F |/Fc = 0.2, 0.6, 1 in

the upper panel and |F |/Fc = 2, 6, 10 in the lower panel.

The current is in units of d2I1(q)

dq2dCg

dyq0Vgyc, which assuming

Vg/Vg0 10−2 amounts to 10−3Ic for the parameter setchosen.

function of time, ϕ = ωjt, ωj = 2eV/� being the Joseph-son frequency which corresponds to the voltage V acrossthe junction. In the same approximation, the current is apurely oscillatory function of time. The time-dependentcurrent can be expanded into harmonics of the Josephsonfrequency,

I(t) =2e

�

∂Ej

∂ϕ(q, ϕ = ωjt) =

∑n=1

In sin(nωjt). (40)

A d.c. current emerges from the feedback on Joseph-son dynamics: oscillatory currents produce oscillatorycorrections to the phase. These are proportional to theimpedance at frequencies nωj . Taking this into accountin the first approximation in Z(ω), we arrive at

Id.c. =∑n

|In|2ReZ(nωj)

V. (41)

The above relation holds provided ReZ � V/Ic. Non-perturbative treatment of Josephson dynamics is re-quired otherwise.Let us consider the mechanical effects. It is important

to note that under the d.c. voltage bias the Josephsonforce also oscillates in time,

Fj(t) = − dCg

dyVg0

∂Ej(q, ϕ)

∂q.

= − dCg

dyVg0

∞∑n=1

∂Ej,n(q)

∂qcos(nωjt) , (42)

Ej,n being the harmonics of Josephson energy. Therefore,the force can efficiently excite the mechanical resonatorprovided nωj � ω0. Let us first concentrate on the casewhere the resonance frequency is matched by the firstharmonics, ωj � ω0. The detuning is defined as ν =ωj − ω0.To start with, let us assume that the Josephson force

is sufficiently weak so that the mechanical response islinear and thus, given by Eq. (35). The direct mechanicalcontribution to the d.c. Josephson current is obtained byaveraging Eq. (27), and reads

Im =∂I1

∂(q/e)

(dCg

dy

Vg

e

)Imy (43)

This can be cast to the form similar to (41),

Im =

∣∣∣∣ ∂I1∂(q/e)

∣∣∣∣2ReZm(ωj)

V(44)

where the current is replaced with detecting current∂I/∂(q/e) and the ”mechanical impedance” Zm(ω) is de-fined as

Zm(ν) =ω0

−iν + Γ/2Z(0)m ; (45)

Z(0)m ≡ �

e2

(dCg

dy

Vg

e

)2�

2Mω0. (46)

(Here, ν ≡ ωj − ω0).This form of the mechanical response makes evident

an analogy with Fiske steps25 that are observed at volt-ages corresponding to resonant frequencies of an electri-cal impedance. This may be either an impedance of ex-ternal circuit or an effective impedance that is essentiallycontributed Josephson inductance.To comprehend the scale of the response, we note first

that for a sufficiently well-developed Coulomb blockadeI1 � ∂I1/∂(q/e). Therefore, to compare the currentgiven by Eq. (41) and the mechanical response, we needto compare Zm and a typical environmental impedance.The latter can be estimated as the impedance of freespace Zf � 102 Ohm. The typical mechanical impedance

far from the resonance, Z(0)m , should be much smaller

than that. Indeed, if we substitute the parameter set(Eq. (29) in Subsection III F) into Eq. (46) we end up

with Z(0)m = 0.7 · 10−2 Ohm � Zf . However, Zm is en-

hanced by a factor of Q at the resonant frequency. Withthis, Zm > Zf and the current peak produced by the me-chanical response should exceed the background currentgiven by Eq. (41) and be clearly observable.The voltage dependence of d.c. current response in

linear regime is determined by ReZm and thus takes aLorentzian shape with the half-width δV = V/Q. Thisassumes a noiseless voltage source. It is known23 thatthe voltage noise suppresses the coherence of Josephsongeneration. For white noise spectrum of intensity SV , theresulting line-width reads δVn = (2e/�)2SV . Comparing

225


12

FIG. 5: (Color online.) Top left panel: the charge-dependentpart of Josephson energy as a function of the gate-inducedcharge q = CgVg. The crosses indicate the values of q thatcorrespond to the values of Josephson force used in other pan-els. The blue dotted line indicates the value of Ej whereFj = Fc. Top right panel: the mechanical response as afunction of detuning for relatively low values of the forceFj/Fc < 1, at which the response increases with increasingthe force. Bottom panels: frequency dependence at the forcevalues Fj/Fc = 3, 5, 7, 10 (from the left to the right panel)where the response decreases with increasing the force.

the two, we conclude that the mechanical response willbe broadened by the noise and essentially suppressed pro-vided δVn > δV , this is, SV > (2e/�)ωj/Q. Detectionof the mechanical response requires (2e/�)ωj/Q � SV ,a condition that may be challenging to meet in practicalcircumstances.As mentioned, the Josephson force can be big enough

to exceed Fc, this makes it relevant to address the non-linear response as well. We illustrate the non-linear re-sponse in Fig. 5. In order to produce this Figure we tookthe charge-dependent part of the Josephson energy to beof the form Ej(q, ϕ) = E cos(ϕ) cos(πq/e). Tuning q withthe d.c. gate voltage tunes the magnitude of the Joseph-son force from 0 to a maximum value π(q/e)(E/Lg). Forillustration we chose E such that the maximum forceπ(q/e)(E/Lg) = 10Fc (E = 2.3 μeV for the parame-ter set in use, corresponding to a 30% charge modulationof the total Josephson energy) and compute the responseusing Eq. (43) and Eq. (5) at a set of the values of q,or, equivalently, Fj . The response is Lorentzian at smallforces, increases and develops a jump characteristic forbistability. It is interesting to note that the responseslowly decreases upon increasing Fj at Fj > 2Fc. This isbecause the response is proportional to Imy that quickly

decreases at big driving forces. In this limit, Im ∝ F−1/3j .

A. Excitation by higher harmonics

If we take higher harmonics of current-phase character-istic into account, we note that Josephson force emerges

at a set of frequencies that are integer multipliers of ωj

(Eq. (42)). This implies that the resonant mechanicalresponse can be also observed in the vicinities of a set ofdiscrete voltage values satisfying ωj = ω0/n, this is, atlower voltages than the resonance described above. Theresponse is computed along the same lines with replacingEj,1 by Ej,n. In linear regime, the response reads

Im =

∣∣∣∣ ∂In∂(q/e)

∣∣∣∣2nReZm(nωj)

V(47)

(cf. Eq. 44, the factor n in the present expression iscanceled by lower voltage V = (�/2e)ω0/n) The responsescales with the relative values of the harmonics and is inprinciple of the same order of magnitude for several lowharmonics. Its dependence on voltage in the vicinity ofthe resonance is similar to that discussed above and doesnot have to be illustrated separately.

B. Parametric excitation

For the sake of completeness, let us mention the pos-sibility of the resonant mechanical response at highervoltages by means of parametric excitation12. Generally,parametric resonance in a non-linear oscillator is achievedby applying an a.c. driving force with frequency abouta double of the resonant frequency, Ω � 2ω0

12. In ourcase, this is achieved by applying a d.c. bias voltage withωj � 2ω0, so that the Josephson force oscillates at 2ω0

and integer multiples of this frequency and thus providesthe parametric driving required.The response of at resonant frequency emerges pro-

vided the parametric driving force exceeds a certainthreshold value, and, as in case of direct resonance,achieves values � yc. The point is that this thresh-old driving force is parametrically bigger than Fc, Ft �Fc

√Q. For our devices at Q = 105, the parametric ex-

citation requires Josephson energies that by a factor of30 exceed the value from the parameter set and are notpractical. This is why we do not explore the regime ofparametric excitation in detail.Besides, the manifestation of the oscillating amplitude

is not as straightforward as in the case of direct reso-nance. The contribution of displacement at ω0 to themechanical current response (27) oscillates at the samefrequency and is not readily rectified to a d.c. current.Under our assumptions, the d.c. mechanical responseis dominated by the displacement oscillating at 2ω0 andis by a factor of

√Q smaller than the typical responses

studied in this paper.

VI. SHAPIRO STEPS AT RESONANT DRIVING

From now on, we turn to the situation where the setupis a.c. driven at frequency Ω. As discussed in SectionIIA, in our setup this gives rise to two a.c. signals Vg(t) =

226

C.VI. Shapiro steps at resonant driving

13

Vg cos(Ωt + χ), Vb(t) = Vb cos(Ωt). The effect of Vb is aformation of Shapiro steps26.A common approach to Shapiro steps takes into ac-

count only the first harmonics of the current-phase re-lation and starts with the assumption that the time-dependent superconducting phase difference can be pre-sented as a sum of three terms

ϕ(t) = ϕ1 sin(Ωt) + ωjt+ ϕ0. (48)

Here, the first term describes the a.c. driving (ϕ1 > 0,

ϕ1 = |Vb|/(2e/�)Ω, the second term corresponds to a d.c.voltage V = ωj/(2e/�), and the third term is a lock-inphase important for further consideration. With this,sin(ϕ) can be presented as a sum over harmonics

sin(ϕ) =

∞∑m=−∞

Jm(ϕ1) sin(Ωmt+ ϕ0) (49)

with Ωm = mΩ + ωj. Here, Jm denotes the m-th Besselfunction of the first kind.Shapiro steps are formed at discrete values of d.c. volt-

age |ωj| = mΩ. In this case, the time-dependent currentI(t) = Ic sin(ϕ(t)) has a d.c. component

Idc = −Icsgn(V )Jm(ϕ1) sinϕ0 (50)

Simplest assumption is ideal current bias at zero fre-quency and ideal voltage bias at frequencies � Ω. Inthis case, the I-V curve of a.c. driven junction consistsof a series of separate pieces. At each piece (Shapirostep) the voltage is locked to one of the discrete values.The current within each piece may vary from minimumvalues I− to the maximum value I+ provided the biascurrent fits this interval. In this case, the actual valueof the lock-in phase ϕ0 is set by the bias current. Theextremal values I± = ±Ic|Jm(ϕ1)| are achieved at thelock-in phases given by

ϕ±0 = ∓π/2 sgn (V Jm(ϕ1)) (51)

We in main follow this approach while admitting ex-treme simplifications it brings. The higher harmonics ofcurrent-phase relation and/or non-ideal voltage bias notonly modify the relation between the current and lock-inphase: They also provide phase-locking at fractional ra-tios of ωj/Ω

28 and formally at all rational values of thisratio. These fractional Shapiro steps are however moresensitive to noise than the integer ones and more likelyto vanish. The I-V curves of our devices do show well-developed steps at integer values of ωj and only tracesof phase-locking at intermediate values. For this reason,we do not consider fractional Shapiro steps in this paperand concentrate on integer ones where |ωj| = mΩ.It is advantageous to look at the mechanical response

at Shapiro steps rather than at d.c. bias conditions. Theexternal a.c. driving synchronizes Josephson oscillations.The inductive response present at Shapiro steps also re-duces significantly the voltage noise at low frequencies sothat it does not broaden the resonant lines.

In this Section, we will consider the mechanical re-sponse in the simplest situation of resonant driving wherethe driving frequency matches the resonant frequency,Ω � ω0.

A. First step

Let us first concentrate on the first Shapiro step, theone at voltage 2eV/� = ±Ω, that is the biggest in thelimit of small driving voltages ϕ1 � 1, and determine thed.c. part of the response at the oscillating displacementy. To represent the results, we normalize y to the non-linearity scale yc and introduce a convenient current scale

I =∂I1∂q

dCg

dyVg0yc (52)

For the values of our parameter set,

I � Ic(q0/e)(yc/Lg) = 2.1× 10−3 Ic.

Making use of Eqs. (49) and (27), we express the me-chanical response in terms of the amplitudes y and Ij atthe resonant frequency,

Im =I

ycRe {j∗y} (53)

j = − i

{ (J2(ϕ1)e

iϕ0 − J0(ϕ1)e−iϕ0

)if V > 0(

J0(ϕ1)eiϕ0 − J2(ϕ1)e

−iϕ0)

if V < 0

(54)

This displacement is a response on the force at resonantfrequency which is a sum of Josephson force and gateforce. The time-dependent Josephson force is expandedin harmonics in the form

Fj(ϕ) = Fj cos(ϕ(t)), (55)

= Fj

∞∑m=−∞

Jm(ϕ1) cos(Ωmt+ ϕ0)

Fj = − dCg

dyVg0

∂E1,j(q)

∂q� q0

e

Ej

Lg.

Its amplitude at resonant frequency is contributed by theterms m = 0, 2 and reads

Fj = F f ; (56)

f =

{ (J2(ϕ1)e

iϕ0 + J0(ϕ1)e−iϕ0

)if V > 0(

J0(ϕ1)eiϕ0 + J2(ϕ1)e

−iϕ0)

if V < 0(57)

Let us discuss first the relative scale of the gate force incomparison with the Josephson force. It may seem thereis none, and varying the a.c. gate voltage Vg one can

achieve any ratio � Vg/EJ between the forces. However,we should take into account the fact that in our setupsa.c. driving also induces an appreciable bias voltage Vb.If the oscillating phase ϕ1 produced by this voltage be-comes large Shapiro steps can hardly be observed. It is

227


14

in general reasonable to expect Vg � Vb. In this case,ϕ1 � 1 corresponds to Fg/Fj � �ω0/Ej . The latter ratiois typically 10−2 in our setups (for our parameter set it is�ω0/Ej = 2.7×10−2). This implies that typically we candisregard a.c. gate force in comparison to the Josephsonforce. We will analyze this case first and consider theeffect of the gate force in the end of the Subsection.With this, the mechanical response is given by

Im =IF

FcRe{j∗fR} =

IF

Fc

(sgnV

(J20 (ϕ1)− J2

2 (ϕ1))Im{R}

− J0(ϕ1)J2(ϕ1) sin(2ϕ0)Re{R}) (58)

Here, R ≡ R(ν/(Γ/2), (Fj/Fc)|f |) defined by Eq. (5)gives the non-linear mechanical response. The expressionis naturally separated onto two terms. The first term isproportional to Im(R) and therefore exhibits a Lorentz-like dependence on frequency. It does not depend on thelock-in phase and can be regarded as a shift in the cur-rent. Owing to the shift, the maximum and minimumcurrents I± at the step are no more opposite: the me-chanical effect breaks the symmetry of the Shapiro step.The shift is however opposite for opposite voltages. Theorigin of the shift may be traced to the Fiske response(Eq. 44) formed at ωj � ω0 in the absence of the a.c.driving. Indeed, in the limit of vanishing ϕ1 the firstterm in the mechanical response does not vanish: rather,it approaches the expression (44). So it looks like theFiske response persists also for well-developed Shapirosteps and contributes to the current at the step. Thissuggest perhaps the easiest way to observe and identifythe mechanical response: measure maximal and minimalcurrents at a step as function of the a.c. frequency. Inthe rest of the paper we thus mainly concentrate on themodification of extremum currents.The second term in Eq. (58) cannot however be ob-

served in this way. In ideal current bias conditions atlow frequency, the second term in the current responsefact amounts to a shift of the lock-in phase. Indeed,since the current at the step as function of ϕ0 reads asI(ϕ) = −sgn(V )J1(ϕ1), the second term can be seen as amodification of the lock-in phase at constant bias currentwhich does not depend on this current,

(Δϕ0)m = −sgn(V )IF

IcFc

J0(ϕ1)J2(ϕ1)

J1(ϕ1)Re{R} (59)

This response is of Fano-type. Since such shift of thephase does not modify the values of the current ex-trema, the effect cannot be observed in the course oftwo-terminal electrical measurement in our setup. Theshift of the lock-in phase can be however revealed if theJosephson junction under consideration is a part of aSQUID, or with the aid of lock-in measurement at non-resonant a.c. frequency.With respect to this, we ought to mention yet another

effect of Josephson force manifesting itself in the me-chanical response considered. In fact, the situation at a

Shapiro step is similar to the phase bias conditions con-sidered in Section IV, with lock-in phase playing the roleof ϕ. We thus expect ϕ0-dependent shift of the resonancefrequency. The static Josephson force at a Shapiro stepis given by

Fj = F sgn(V )J1(ϕ) cos(ϕ0) (60)

The frequency shift caused by this force thus reads

Δω0(ϕ0) = sgn(V )(Δω0)maxJ1(ϕ) cos(ϕ0). (61)

Here (Δω0)max is a maximum frequency shift in the ab-sence of the a.c. driving, given by Eq. (31). The fre-quency shift vanishes at extremum points ϕ±0 and there-fore cannot be observed by measuring the extrema of thecurrent.We illustrate the mechanical response in Fig. 6. In

this Figure as well in all subsequent Figures except Fig.7, we concentrate on the modification of the maximumcurrent on the step. Instead of presenting the (rathertrivial Lorentz-like) frequency dependence of this modi-fication, we give the extremum of this modification overthe frequency range and plot it versus ϕ1. The extremumis proportional to the maximum of Im(R) over the fre-quency. We shall note that the dependence of this max-imum on the force is rather specific one: it is a constantuntil the bistability threshold F = 1.24Fc, has a cusp atthis value of force, and decreases monotonously at higherforces. This accounts for rather strange appearance ofthe response curves. If the Josephson force is smallerthan the bistability threshold, they coincide with the lin-ear response given by the dotted curves. Otherwise, theresponse is smaller than linear one and exhibits kinks.The so-defined maximum response is plotted in Fig.

6 for two values of F , those correspond to slightly andstrongly non-linear regime, respectively. In both cases,the response vanishes when the width of Shapiro stepreaches maximum, or becomes zero (except ϕ1 = 0).In slightly non-linear regime, the response reaches maxi-mum value at ϕ0. Upon increasing ϕ1, it exhibits Bessel-like oscillations. The envelop of these oscillations shrinkswith increasing ϕ1. This shrinking is much faster thanthat for either step width or Josephson force. In stronglynon-linear regime, the amplitude of the response is deter-mined by competition of two factors: it is increased bythe bigger value of the detecting current, and decreasedowing to smaller imaginary part of oscillating displace-ment at higher Josephson forces.Let us turn to the effect of the gate force. The Joseph-

son force of the kind considered can change sign andtherefore be tuned to zero by tuning q. In the vicinity ofthis particular q, the gate force should compete with theJosephson one and eventually dominate. The full ampli-tude of the force at the resonant frequency then reads

F = F f + Fg exp(−iχ), (62)

the frequency shift χ between the bias and the gate volt-age being a relevant parameter.

228

C.VI. Shapiro steps at resonant driving

15

FIG. 6: (Color online.) The mechanical response at resonantdriving Ω ω0 and at the first Shapiro step V0 = 2eΩ/�versus the oscillating phase ϕ1. The first (upper) plot givesthe maximum current at the step. The second and fourthplots give the mechanical response defined as the extremum ofthe modification of this maximum current over the frequenciesnear the resonance, at maximum Josephson forces F = Fc andF = 50Fc, respectively. The actual amplitudes of the resonantJosephson forces are given at the lower plots, respectivelythird and fifth.

The mechanical response is given by Eq. (58) where Rdepends on the full force plus an addition proportional

FIG. 7: (Color online.) The effect of gate force. For allplots, the gate force is fixed to Fg = Fc. In the plots fromtop to bottom the maximum Josephson force F assumes thevalues F /Fc = −5,−1, 0, 1, 5. We choose χ = 0 and ϕ1 = 1.Dashed lines in the plots for I− give values opposite to thecorresponding I+, to stress the symmetry or asymmetry ofthe response.

229


16

to Fg,

I(g)m =IFg

FcRe{j∗ exp(−iχ)R} =

IF

Fc(−(J0(ϕ1) sin(ϕ0 + χ) + J2(ϕ1) sin(ϕ0 − χ))Re(R)

(J0(ϕ1) cos(ϕ0 + χ)− J2(ϕ1) cos(ϕ0 − χ))Im(R))(63)

The last equation holds for V > 0. The expression forV < 0 is obtained by interchanging J0 and J2. Evaluat-ing this at the extremum points of lock-in phase, ϕ±0 , weobtain

I(g)±m = ∓ sgn(V J1(ϕ1))IFg

Fc(64)

(J0(ϕ1) + J2(ϕ1))Re{R exp(−iχ)}

Therefore, the contribution of the gate force to extremumcurrents is not like a shifts: rather, it modifies the widthof the step. These terms are even in voltage and displaya mixture of Fano-type and Lorentzian-type response asfunction of frequency, this being tuned by the phase χ.To illustrate a rather complex interplay of Josephson

and Shapiro dynamics we plot in Fig. (7) the frequencydependence of the mechanical response for a constantVg and a set of values of Fj that pass zero. The plotsshow the modifications of extremum currents I±. Thesemodifications are the same for the Josephson force con-tribution and opposite for the gate force contribution.Besides, the frequency dependence is Fano-like for thegate force contribution and Lorenz-like for the Joseph-son force contribution. In the central plot, the Joseph-son force contribution is absent, the modifications of I±

are opposite, and the frequency dependence is Fano-like.Upon increasing the Josephson force, these features aretransformed into the opposite ones. The plots are sym-metric upon simultaneous change of signs of the currentand the Josephson force.

B. Higher steps

At the same conditions of the resonant driving, we ana-lyze the mechanical response at other Shapiro steps m >1, those correspond to higher voltages |V | = m�ω0/2e.Both the amplitudes of the detecting current and theJosephson force display a complex dependence on thestep number m and the oscillating phase ϕ1. They aregiven by

j = −i(J−1+meiϕ0 − J1+me−iϕ0

)f = J−1+meiϕ0 + J1+me−iϕ0

where the dependence on the sign of the voltage is in-corporated into m ≡ −sgn(V )m. We consider only thesituation when Josephson force dominates. With this, we

obtain a relation similar to Eq. (58):

Im =IF

FcRe(j∗fR) =

IF

Fc

(sgnV

(J2m−1(ϕ1)− J2

m+1(ϕ1))Im(R)

− Jm−1(ϕ1)Jm+1(ϕ1) sin(2ϕ0)Re(R)) (65)

As in the first step, the response consists of two terms.The first one gives a shift in the current, and gives amodification of the maximum and minimum currents atthe step, this is to be measured. As in the previous case,the shift is odd in voltage. However, its ϕ1-dependenceis quite rather distinct.

The measuring of the mechanical response at highersteps is important to check the consistency of results andthereby unambiguously identify the mechanism of the re-sponse. The characteristic dependences on ϕ1 make theidentification easy.

We illustrate the response for higher steps in Fig. 8(second step) and Fig. VIB (fifth step). In both cases,the response correlates with the Shapiro step width givenin the upper plots: it vanishes when the width achieves amaximum or becomes zero. In distinction from the firststep, the responses vanish at vanishing ϕ1. Their typicalvalues are of the same order. However, the envelopes ofthe responses decrease rather slow with increasing ϕ1.

VII. SHAPIRO STEPS AT NON-RESONANTDRIVING

In the previous Section, we concentrate on the casewhen the driving frequency Ω matches the resonant fre-quency of the mechanical oscillator. It is not a neces-sary condition for an efficient excitation of the resonantmode. The Josephson dynamics at Shapiro steps are es-sentially non-linear. As a consequence, the spectrum ofcurrent oscillations contain all higher harmonics nΩ ofthe driving frequency Ω. The same pertains the Joseph-son force. Therefore, the resonator can be efficiently ex-cited for Ω = ω0/N , N > 1 being an integer number.At any given N , the resonant conditions are achievedfor any Shapiro step number m, and thus for voltages2eV/� = ωj = (m/N)ω0.

These non-resonant driving conditions are advanta-geous for observation of the Josephson force since thea.c. gate voltage force is not in the resonance, does notcause any appreciable displacement and therefore doesnot mask the effect of the Josephson force. In this shortSection, we will thus concentrate on the case of the non-resonant driving Ω = ω0/N .

The amplitudes of the detecting current and theJosephson force depended not only on the step numberm and the oscillating phase ϕ1, but also on N . They are

230

C.VII. Shapiro steps at non-resonant driving

17

FIG. 8: (Color online.) The maximum current at the Shapirostep, the mechanical response and the force versus ϕ1 for F =Fc and F = 50Fc at the second Shapiro step V0 = 4eΩ/�.

given by

j = −i(J−N+meiϕ0 − JN+me−iϕ0

)f = J−N+meiϕ0 + JN+me−iϕ0

where the dependence on the sign of the voltage is againincorporated into m ≡ −sgn(V )m.

Since the gate force is absent, the response is given bya relation similar to Eq. 58 that contains the Josephson

FIG. 9: (Color online.) The same as in Fig. 8 at the fifthShapiro step V0 = 10eΩ/�.

force only:

Im =IF

FcRe(j∗fR) =

IF

Fc

(sgnV

(J2m−N (ϕ1)− J2

m+N (ϕ1))Im(R)

− Jm−N(ϕ1)Jm+N (ϕ1) sin(2ϕ0)Re(R)) . (66)

It is again separated onto two terms discussed above,only the first term being responsible for the modificationof the extremum currents of the Shapiro steps.

231


18

FIG. 10: (Color online.) The mechanical response at the non-resonant driving. Here, the a.c. driving frequency is Ω ω0/2,corresponding to N = 2. From left to right the three columns correspond to Shapiro steps m = 1, 2, 3. Plotted are the maximumcurrent at the Shapiro step, the mechanical response and the amplitude of the force. The maximum of the Josephson force wasset to F = 50Fc for all plots.

FIG. 11: (Color online.) The mechanical response at the non-resonant driving for N = 3. Except this, all other parametersare the same as in Fig. 10

We illustrate the dependences on ϕ1 in Fig. 10 (forN = 2) and Fig. 11 (for N = 3) for the first three stepswith m = 1, 2, 3. The vertical arrangement of the plotsis the same as in the previous Figures except we chose asingle value of the maximum Josephson force F = 50Fc

that brings us deep into the non-linear regime. In thisregime, the response is of the same order of magnitude forall steps and ratios N , while retaining unique m,N spe-cific dependence on ϕ that can be used for identificationof the effect and the characterization of the Josephsonforce.

VIII. CONCLUSIONS

In conclusion, we have studied Josephson junction dy-namics affected by excitation of a mechanical resonator.We have demonstrated that the mechanical oscillationscan be rectified giving rise to an additional d.c. currentthat can be used for detection. The mechanical responseis proportional to the oscillation amplitude, and is esti-mated as Im � Ic(q0/e)(y/L) � 10−3Ic(y/yc). The res-onator can be driven by the a.c. voltage applied to thegate electrode as well as an additional mechanical force,termed the Josephson force, that depends on the super-

232

C.A. Lagrangian formalism

19

conducting phase difference at the junction. We estimatethe Josephson force as Fj � (q0/e)(Ej/Lg) and show thatit is sufficiently strong to drive the mechanical resonatorinto the non-linear regime. We also show that it is typi-cally larger than magneto-induced force proposed in21.

We have presented a general and detailed analysis ofthe coupling between electrical and mechanical degrees offreedom, discussing the competing non-linearities. Thisanalysis is applied to a Josephson device with a sus-pended CNT resonator, where we show that the intrinsicnon-linearity scales dominate those arising from the cou-pling.

We have provided analytical formulas for the responseof the device to mechanical excitations in a wide inter-val of the excitation strengths and for various biasingschemes. We discuss distinct frequency dependencies,Lorentz and Fano-like, of the mechanical response bothfor linear and non-linear regimes and show how thesearise based on the nature of the resonant mechanicalforce. In the case of a phase biased junction we showthat the resonant frequency of the mechanical mode ac-quires a measurable phase-dependent shift (see Fig. 3).

We have discussed conditions of detecting the en-hanced mechanical response arising when the Josephsonfrequency matches the resonance frequency of the me-chanical mode. We reasoned that the regime of Shapirosteps is advantageous, since the fluctuations of the volt-age drop over the junction are suppressed. We providedexpressions for the mechanical response in the regime ofShapiro steps and demonstrated that it manifests as mod-ifications of the extrema of the steps. We show that themechanical mode can be efficiently excited not only byresonant a.c. signals, but also by a.c. signals with fre-quencies close to an integer fraction of the mechanical res-onance frequency. Our preliminary experimental resultsconfirm this behavior; these will be reported elsewhere.

Acknowledgments

The authors would like to thank G. A. Steele, S. M.Frolov, L. P. Kouwenhoven, and H. S. J. van der Zant formany useful discussions. This research was supported bythe Dutch Science Foundation NWO/FOM.

Appendix A: Lagrangian formalism

Here we present a derivation of the equations of mo-tion using the Langrangian of our junction model, de-scribing the dynamics of the three variables y, q and ϕ,treated here as generalized degrees of freedom. To ac-count for effects of dissipation we introduce generalizedfriction forces for each degree of freedom and treat theseas external forces.

The kinetic energy of the system is that of the mechan-

ical resonator.

T =1

2My2 (A1)

The inertial effects associated with variations of ϕ, in themain text we have conveniently treated those using theexternal impedance Ze, which describes both inertial ef-fects as well as dissipation. We do the same here treatingboth effects as a generalized external force proportionalto Ze.The potential energy includes contributions from the

mechanical, electrostatic and Josephson energies U =Um + Ue + Ej .

Um =1

2Mω2

0y2 −M

α

3y3 −M

β

4y4; (A2)

Ue = Ec(q) +q2

2Cg(y)− qVg (A3)

The system Lagrangian is given by L = T −U . Equa-tions of motion are obtained using:

d

dt

(∂T

∂x

)− ∂T

∂x= −∂U

∂x+ Fx; (A4)

where x = y, g, ϕ denotes a generalized degree of freedomand Fx is the corresponding generalized friction force.The mechanical friction force is modeled typically as

linear in variation of displacement Fy = −MΓy.Dissipative effects associated to charging of the gate

capacitance are important on timescale RgCg, where Rg

is the negligible resistance of wires grounding the gateand source-drain electrodes. We assume that RgCg �ω−10 , such that the effect of gate charging on the dynamics

of y and ϕ can be safely neglected Fq = 0.The generalized friction force describing fluctuations

of the phase corresponds to the current passing throughthe circuit impedance Ze, that is

Fϕ =�2

4e2

∫ t

−∞dt′dt′′Z−1

e (t′, t′′)(Vb − ϕ(t′′)). (A5)

With this, the equations of motion are:

My =− ∂U

∂y−MΓy (A6)

0 =− ∂U

∂q(A7)

0 =− ∂Ej

∂ϕ+

�2

4e2

∫ t

−∞dt′dt′′Z−1

e (t′, t′′)(Vb − ϕ(t′′))

(A8)

equivalent to those presented in the main text.The equivalent Hamiltonian is

H = T (p) + U , T (p) = p2/2M ,

where p is the momentum of the resonator p = My.There is no generalized momentum associated to ϕ and

233


20

q, since their inertial effects are not included in the La-grangian.Using the Hamiltonian, we obtain the same equations

of motion using

p =− ∂H∂y

+ Fy (A9)

0 =− ∂H∂q

(A10)

0 =− ∂H∂ϕ

+ Fϕ (A11)

234

Bibliography

21

1 A. N. Cleland, Foundations of Nanomechanics (Springer-Verlag, Berlin, 2003).

2 K. L. Ekinci, X. M. H. Huang, and M. L. Roukes, Appl.Phys. Lett. 84, 4469 (2004).

3 M. P. Blencowe and E. Buks, Phys. Rev. B 76, 014511(2007).

4 A. N. Cleland and M. L. Roukes, Nature (London) 392,160 (1998).

5 A. D. Armour, M. P. Blencowe, and K. C. Schwab, Phys.Rev. Lett. 88, 148301 (2002).

6 T. L. Schmidt, K. Børkje, C. Bruder, and B. Trauzettel,Phys. Rev. Lett. 104, 177205 (2010).

7 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, A.N. Cleland, Nature(London) 464, 697 (2010).

8 I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippen-berg, Phys. Rev. Lett. 99, 093901 (2007); F. Marquardt, J.P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett.99, 093902 (2007).

9 G. Sonne, M. E. Pena-Aza, L. Y. Gorelik, R. I. Shekhter,and M. Jonson, Phys. Rev. Lett. 104, 226802 (2010).

10 M. D. LaHaye, J. Suh, P. M. Echternach, K. C. Schwab,and M. L. Roukes, Nature (London) 459, 960 (2009).

11 H. G. Craighead, Science 290, 1532 (2000); M. Blencowe,Phys. Rep. 395, 159, (2004).; K. L. Ekinci, and M. L.Roukes, Rev. Sci. Instrum. 76, 061101 (2005).

12 J. Guckenheimer and P. Holmes, Nonlinear Oscillations,Dynamical Systems, and Bifurcations of Vector Fields(Springer-Verlag, New York, 1983).

13 V. Sazonova, Y. Yaish, H. Ustunel, D. Roundy, T.A. Arias,and P. L. McEuen, Nature (London) 431, 284 (2004).

14 B. Witkamp, M. Poot, and H. S. J. van der Zant, Nano.

Lett. 6, 2904 (2006).15 G. A. Steele, A. K. Huttel, B. Witkamp, M. Poot, H. B.

Meerwaldt, L. P. Kouwenhoven, and H. S. J. van der Zant,Science 325, 1103 (2009).

16 A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos,and M.M.J. Treacy, Phys. Rev. B 58, 14013 (1998).

17 A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A.Clerk, M. P. Blencowe, and K. C. Schwab, Nature (Lon-don) 443, 193 (2006).

18 In more rigorous terms, I is the moment of inertia per massdensity per unit length of the cross section.

19 L. D. Landau and E. M. Lifshitz, Theory of Elasticity(Pergamon, New York, 1986).

20 S. Sapmaz, Y. M. Blanter, L. Gurevich, and H. S. J. vander Zant, Phys. Rev. B 67, 235414 (2003).

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22 In this case, the critical current is estimated from theproduct of the junction conductance and superconductingenergy gap and can exceed the experimentally measuredswitching current by two orders of magnitude.

23 K. K. Likharev, Dynamics of Josephson Junctions and Cir-cuits (Gordon and Breach, New York, 1986).

24 Y. V. Nazarov and Y. M. Blanter, Quantum Transport:Introduction to Nanoscience (Cambridge University Press,Cambridge, 2009).

25 M. D. Fiske, Rev. Mod. Phys. 36, 221 (1964); D. D. Coonand M. D. Fiske, Phys. Rev. 138, A744 (1965).

26 S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).27 C. J. H. Keijzers, L. P. Kouwenhoven, unpublished.28 P. W. Anderson and A.H. Dayem, Phys. Rev. Lett. 13, 195

(1964).

235

Appendix D

Characterization of rheniumfilms

In this appendix we describe the characterization of the rhenium films that weredeveloped for the experiments in this thesis. This chapter has been adapted fromthe master thesis of Vincent Mourik [1].

D.1 Experimental goal

Experiments on Re film deposition and characterization are presented. The goalof these experiments was twofold. On the one hand, the compatibility with theCVD CNT growth procedure of thin sputtered Re films is investigated. Theother goal was to characterize such films both at room temperature and lowtemperature.

D.2 Methods

We prepared three different films. The sputtering details can be found in App. E.Here we present the results on the characterization of these films. The three filmsdiffer in the sputtering time used, resulting in different film thickness.

In all cases, the film thickness is measured using the α-stepper (reflectometry)and marker-pen lithography.

Besides the film thickness d, also the sheet resistance Rs of the film is required tocalculate the resistivity ρ of the film (which is ρ = d×Rs). The sheet resistanceis extracted with the Van der Pauw (VdP) method [2]. We used this method intwo different ways.

First a Keithley digital multimeter with a four point prober (equidistant probes)is used to measure the sheet resistance. In the VdP method it is assumed thatthe measurement is done on an infinite film plane. The sheet resistance can then

237

D. Characterization of rhenium films

Table D.1: Room temperature properties of our Re films. Standard deviation for eachnumber added in brackets.

tsp thickness growth rate ρ after sputtering ρ after RTA ρ after CVD(s) (nm) (nm/s) (μΩ · cm) (μΩ · cm) (μΩ · cm)

28(1) 21(3) 0.8(0.1) 65(3) 52(3) 111(6)

40(1) 42(3) 0.83(0.05) 72(4)54(3) 47(2)

No RTA 64(3)100(1) 105(20) 1.1(0.2) 112(21) 75(12) 56(3)

be obtained by multiplying the measured resistance by an factor of π/ ln(2).

Secondly, the full VdP method can be used by a four point measurement appliedat the corners of a film. In such a measurement four wires are bonded to the filmat the four corners. By measuring the resistance in two different configurations,two different resistance values can be found which solve the VdP equation forthe sheet resistance of a film, e−πR12,34/Rs + e−πR23,41/Rs = 1. Numbering thewires from one to four, a bias current is applied through the first two indicated(subscripts to R) wires and the corresponding voltage over the other two indicatedwires is measured. This enables the calculation of the corresponding resistance.

The resistivity of the films is measured after different processing steps. The firstmeasurement is done directly after sputtering of the films. We also investigatedthe effect of Rapid Thermal Annealing (RTA) on the film properties. PossiblyRTA will result in a better film quality by decreasing film disorder and/or surfacecontamination. RTA is performed for 10 minutes at 900 �C in Ar. One wafer withfilm (42 nm) is cut in two, one half exposed to RTA, the other not, enabling us toevaluate the effect of RTA after exposition to the growth conditions. After theRTA step, we measure the resistance again. Both sheet resistance measurementsare done by using the equidistant prober.

The films are exposed to the CNT growth conditions (CVD at 900 �C in thepresence of H2 and CH4) before they are cooled down. After the CVD step, thefilms are glued to a chip carrier. Four wires are bonded from every film to the chipcarrier and the chip carrier is mounted in a 3He cryostat. Before cooling downthe films, the resistivity is measured at room temperature. To investigate thelow temperature properties, the films are cooled down to a typical temperatureof 300mK in the 3He cryostat . In this cryostat, an in-plane magnetic field canbe applied to find the in-plane critical field of the films.

238

D.3. Results

Table D.2: Low temperature properties of the Re films. Standard deviation for eachnumber added in brackets.

thickness ρ after CVD ρ at 300mK RRR Tc Bc

(nm) (μΩ · cm) (μΩ · cm) (ρRT/ρ300mK) (K) (T)

21(3) RTA 111(6) 73(4) 1.52(0.08) 2.4(0.1) 2.1(0.1)42(3) RTA 47(2) 25(2) 1.84(0.09) 2.7(0.1) 1.5(0.1)42(3) No RTA 64(3) 71(4) 0.90(0.05) 2.6(0.1) 1.5(0.1)105(20) RTA 56(3) 61(3) 0.91(0.05) 2.8(0.1) 1.2(0.1)

D.3 Results

In Tab. D.1 we summarize the room temperature results of the resistivity mea-surements. It shows that the film resistivity after sputtering is large comparedto the bulk value of 18.7μΩ · cm. This means that the films are most likely dis-ordered and/or dirty. RTA lowers the resistivity but only to a limited extend, asis shown by comparison to the 42 nm case. The data shows clearly that the CNTgrowth procedure strongly influences the film properties by decreasing (42 nmand 105 nm) or increasing (21 nm) the resistivity, making the RTA procedurealmost insignificant.

In Fig. D.1 and Tab. D.2 we summarize the results of the low temperature charac-terization of our films. We find that the critical temperature of the films is largecompared to the bulk value of 1.7K. Such an increase in Tc of a thin, disorderedand/or dirty superconducting film compared to the bulk Tc is observed beforefor various different metals and various growth techniques [3–5]. Its origin is notfully understood. The value of Tc is far above typical dilution fridge temper-atures, enabling superconductivity in such a measurement setup. The in-planecritical field of the films has values of 1 . . . 2T, and we find that thinner filmshave a higher critical field.

239

D. Characterization of rhenium films

0 1 2 30

5

10

15

20

25

IBias (μA)

V (m

V)

Rn = 9.41 Ω

1 1.5 2 2.50

200

400

600

B (T)

V (μ

V)

Bc = 2.1 T

2 2.5 3 3.5

0.2

0.4

0.6

T (K)

V(m

V)

Tc = 2.4 K 0

(a)

(b) (c) (d)

Figure D.1: Overview of low temperature results. (a) Low magnification SEMimage of three samples mounted on a chip carrier, showing the four bonding wiresconnected to the chip carrier bonding pads. (b) IV -curve of the 20 nm film. Thetrace is taken at T ∼ 300mK. (c) In-plane magnetic field dependence of the 20 nmfilm conductivity. The applied current bias is 100μA, and the measurement is done atT ∼ 300mK. (d) Temperature dependence of the 20 nm film resistance. The appliedcurrent bias is 100μA.

D.4 Conclusion

Our results show that we can deposit thin Re films on a silicon oxide substrate bysputtering. These films are robust against the CVD growth procedure for CNTs.We have shown that the critical temperature of the films is high enough to serveour purpose of building a suspended CNT resonator with superconducting leads.We find that our films have an in-plane critical field as high as 2.1T.

240

Bibliography

Bibliography

[1] V. Mourik, Nano-Mechanics with Superconducting Suspended Carbon Nan-otubes, MSc. thesis, Delft University of Technology, 2010.

[2] L. van der Pauw, A Method of Measuring Specific Resistivity and Hall Effectof Discs of Arbitrary Shape, Philips Research Reports 13(1), 1 (1958).

[3] B. Abeles, R. Cohen, and G. Cullen, Enhancement of Superconductivity inMetal Films, Physical Review Letters 17(12), 632 (1966).

[4] J. Hauser, Enhancement of Superconductivity in Aluminium Films, PhysicalReview B 3(5), 1611 (1971).

[5] R. Pettit and J. Silcox, Film Structure and Enhanced Superconductivity inEvaporated Aluminium Films, Physical Review B 13(7), 2865 (1976).

241

Appendix E

Fabrication recipes

In this appendix we present the detailed process flow to fabricate the devicesdescribed in this thesis. A more general description of the fabrication process isgiven in Ch. 3. When ultrasound is applied, this is always done at the highestpower available with the sonicator. Resist spinning: 5 s spinning at 500 rpm and55 s at the given spinning speed.

E.1 Fabrication of graphene Josephson junctions

Here we will report the fabrication recipe of vanadium trilayer graphene Joseph-son junctions. We omit the recipe for niobium and niobium trilayer junctions.They can be made following this recipe, but using niobium instead of vanadium.

WaferWe used 4 inch, 500(0.25)μm thickness prime grade Si wafer from NOVA. Thewafer consists out of P++ Si (<100>orientation). The dopant is Boron, resultingin a wafer resistivity of 0.04 . . . 0.06Ωm. The wafer is covered on both sides with285(15) nm thermal oxide. Wafers are spin coated with photo resist and diced in19mm2 chips.

E.1.1 Wafer cleaning

Strip photo resist

� 2’ acetone ultrasonic

� 2’ ethanol ultrasonic

� 2’ H2O ultrasonic

� 2’ HNO3 ultrasonic

� 2’ H2O ultrasonic

243

E. Fabrication recipes

� 2’ IPA ultrasonic

� N2 blowdry

Ionic clean15’ RCA 1 at 70�CRCA 1: HCl/H2O2/H2O=30/30/180 ml

E.1.2 Bitmarker lithography

Spin coat e-beam resistPMMA(17.5%)/MMA, 8% in Ethyl lactate, 60 s at 3000 rpm (thickness∼ 325 nm)bake 10’ at 175�CPMMA 950K, 2% anisole, 60 s at 1500 rpm (thickness ∼ 90 nm)bake 10’ at 175�C

EBPGWe used four bitmarker fields per chip.Layer 1: marpad4jun fine.gpf, beam: 22NM 9150PA 400UM 100KV, dose: 1300μC/cm

2

Layer 2: marpad4jun fine.gpf, beam: 22NM 9150PA 400UM 100KV, dose: 1200μC/cm2

Develop60” MIBK:IPA 1:360” IPA

Deposition25 nm Cr at 1 A/s, P ≈ 6× 10−8 Torr45 nm Au at 1 A/s, P ≈ 6× 10−8 Torr

Lift-offfew hours RT (room temperature) acetone2’ IPA rinseN2 blowdry

E.1.3 Graphene deposition

Wafer cleaningOrganic clean: 10’ RCA 1 at 74�C NH4OH/H2O2/H2O=30/30/150 ml, H2Orinse.Ionic clean: 2’ RCA 2 at 74�C HCl/H2O2/H2O=20/20/120 ml, H2O rinse.N2 blowdry

Mechanical exfoliationI started with a relatively big piece of graphite, and zipped/unzipped the Scotchtape 9 times in total before deposition. Then I push it with a hard force (as hard

244

E.1. Fabrication of graphene Josephson junctions

as I can) on the chip for about 30”.10’ wait

First peel: Peel off the tape very slowly, this can take about 1’ per chip.Glue remove: 10’ acetone at 52�C, with magnet stir.2’ IPA rinseN2 blowdry30’ bake at 120�C

Second peel: Press clean tape with hard force on the chip for about 30”. Thenslowly peel it off.

Microscope inspection: If there is glue, do another glue remove. Otherwise not.

Graphene selectionWe used the optical microscope in the MED lab, for selection of graphene byoptical contrast.

E.1.4 Contact lithography

Spin coat e-beam resistZEP520A:Anisol, 1:1, 2000 rpm, bake 15’ at 175�C, typical thickness: 1550 nm

EBPGFine structure layer: beam: 21NM 544PA 400UM 100KV, spotsize: 3 nm, dose:300μC/cm

2.

Medium structure layer: beam: 21NM 544PA 400UM 100KV, spotsize: 23 nm,dose: 300μC/cm

2.

Develop90” n-amylacetate30” MIBK:IPA 9:1N2 blowdry

AJA depostionWe will deposit trilayer films of Ti/V/Ti with at thickness of 35/100/35 Aby e-beam evaporation. With this recipe you will get a film with a thick-ness of 170(20) A, a sheet resistance R� = 47(3)Ω, and a resistivity of ρ =84.2(6.8)μΩcm (standard deviation between brackets). Bulk resistivity of vana-dium is 19.7μΩcm. The resistance of a typical 1μm × 10μm strip is 390Ω atroom temperature, and 43Ω at 2K. Our films have a residual resistance ratioRRR = 9.1.

Vanadium has a high melting point and requires a large power to evaporate. Onehas to be aware of this, and be careful not to overheat the cooling system of theAJA.

Load chips in main deposition chamber at the end of the day and pump thesystem overnight.

245

We use no additional methods to heatsink the chip during deposition.Background pressure after loading and 16 hours pumping P = 4.4× 10−8 Torr.After 5’ vanadium getter pumping: P = 3.2× 10−8 Torr.

35A TiAfter 3’ titanium getter pumping at 1 A/s: P = 2.8× 10−8 Torr.Current during deposition: Ibeam = 0.2A.Rate during deposition: ΓTi = 1.0± 0.2 A/s.Deposition time for 35 A: ∼ 33”.

100A VNote for vanadium deposition: To get 100 A thickness, you have to evaporateuntil the Inficon gauge reads 140 A.Current during depostion: Ibeam = 0.35A.Rate during deposition: ΓV = 6.0± 0.5 A/s.Inficon settings: Tooling 100, mass 5.93 g/cm3, z-ratio 0.530.

35A TiAfter 3’ titanium getter pumping at 1 A/s: P = 1.5× 10−8 Torr.Current during deposition: Ibeam = 0.2A.Rate during deposition: ΓTi = 1.0± 0.2 A/s.Deposition time for 35 A: ∼ 33”.

Lift off12 hours RT anisole, sample under angle facing the bottom of the beaker.IPA spray and rinse.N2 blowdry.

E.2 Fabrication of suspended carbon nanotubeJosephson junctions

This section has been adapted from the master thesis of Vincent Mourik1 .

E.2.1 Ohmic contacts and trench

WaferSee graphene recipe, Sec. E.1.

Strip photoresistSee graphene recipe, Sec. E.1.1.

1V. Mourik, Nano-Mechanics with Superconducting Suspended Carbon Nanotubes,MSc. thesis, Delft University of Technology, 2010.

E.2. Fabrication of suspended carbon nanotube Josephson junctions

Rhenium sputtering

� We used a rhenium target from Kurt J. Lesker company. It has a purity of99.95%, a diameter of 2 inch and a thickness of 0.25 inch.

� Sputtering: We used an AJA international Inc. evaporation and sputteringsystem. Below we give the sputtering recipe. We use the highest DC powerpossible in our system.

� Sample loading:

– Main chamber pressure start: ∼ 4× 10−9 Torr.

– Prepare small test chip with some permanent marker lines on it. Donot forget the testchip, this chip is needed for the etching procedure!

– Mount chips (maximal 2) and testchip on sample holder (lowerrightcorner).

– Place sample holder in loadlock and pump it down to ∼ 8×10−7 Torr.

– Load sample holder into main chamber.

– Main chamber pressure after 5 min.: ∼ 1.5× 10−8 Torr.

� Presputtering:

– Open Ar valve, flow controller on 15 sccm.

– Reduce opening valve to turbopump with VAT gauge controller. Cham-ber pressure should increase to 2μbar on VAT gauge.

– Open sputtergun shutter.

– Increase DC power slowly (100 W/20 s) from zero up to 505 W (max-imum power).

– Presputter for 2 min.

– Decrease power slowly (100 W/20 s) to zero.

Remarks: During presputtering, the sample is loaded, but not facing thesputtergun, to prevent deposition during sputtering. Care should be takenwhen the main chamber has been opened and/or the target is placed. Insuch a case, a longer presputtering time is advised, to remove surface oxidefrom the target.

� Sample positioning:

– Tilt sample holder -38�.

– Rotate sample holder in-plane 58�.

– Tilt sputtergun to 0.25 inch.

– Lower holder to minimum distance to sputtergun.

� Sputtering:

– Close sputtergun shutter.

– Increase power slowly (100 W/20 s) to 505 W (maximum power).

247


Table E.1: Structure of the dry etchmask. Layers presented in order of deposition.

Layer Spinning speed tbaking Tbaking Thickness(rpm) (s) (�C) (nm)

1 PMMA950K (3% anisole) 6000 90 175 100

2 S1805 7000300 120

4001800 175

3 Tungsten - - - 74 PMMA950K (4% anisole) 4000 90 175 150

– Presputter 10 s with shutter closed.

– Open shutter and sputter for 29 s (this will result in ∼20 nm Re).

– Typical parameters: P = 505W, I = .977A, V = 510V.

– Close shutter, decrease power slowly (100 W/20 s) to zero, open valveturbopump and close Ar line.

– When main chamber pressure reaches ∼1×10−6 Torr, unload samples.

� Re thickness measurement:

– Lift off markerlines testchip with acetone sonica for ∼ 1min.

– Re film thickness with reflectometry (α-stepper).

EtchmaskTab. E.1 summarizes the structure of the etchmask. All important fabricationparameters are given, except for the tungsten evaporation, which is treated belowseparately. It is important to start the fabrication of the etchmask as soon aspossible after sputtering the Re film. In this way, possible adhesion problems canbe prevented. For the same reason, after evaporation of the tungsten layer, thefinal resist layer should be deposited as soon as possible.

� Cleaning Re film:

– 1’ acetone

– 1’ IPA

– N2 blowdry

� Spinning resist layers:

– Spin PMMA950K anisole 3% and S1805 layers with settings given inTab. E.1.

� Tungsten evaporation:

We used a Temescal evaporation system to deposite tungsten. Alterna-tively, an Allianz sputtering system can be used. It is also possible to useGe as an etchmask.

248


– Load chips into Temescal. Include testchip with permanent marker-lines. Do not forget the testchip, this chip is needed for the etchingprocedure!

– Modify tungsten evaporation recipe. Use a typical power of 23% andan aimed thickness of 7 nm.

– Run recipe and unload after finishing.

� W thickness measurement:

– Lift off markerlines testchip in ultrasonic acetone for ∼1 min.

– Measure W film thickness with the α-stepper.

� Spinning final resist layer:

– Spin PMMA950K 4% anisole with the settings given in Tab. E.1.

� Inspection etchmask:

– Make a small scratch at the edge of the chip and measure the totalthickness of the etchmask with the α-stepper. The thickness shouldbe ∼ 700 nm.

– Check the etchmask with optical microscope. Strange colors and/orshapes should not be present.

� Electron beam patterning:

The system used is a Leica 5000+ electron beam pattern generator. Thedesign consists of coarse and fine parts, which are written with a beamsizeof respectively 93 nm and 4 nm. In both cases, the dose is 1200μC/cm

2.

– Mount chip to sample holder.

– Height alignment of the chip has to be within ±5μm of the referenceheight of the sample holder.

– In-plane alignment should be done by using the chip edge as reference(no markers are present yet).

– Use the coordinates of two opposite corners to calculate the centercoordinates.

– No marker search is needed (this is the first patterning step), only thecenter coordinates should be entered in the layout file.

– Load sample holder into EBPG and perform the writing job specifiedby the layout file. After finishing, unload the sample holder. The chipis ready for development.

� Resist development:

– Prepare beakers, one with MIBK:IPA = 1:3 and one with IPA.

– Gently move the chip through the MIBK:IPA = 1:3 for 90 s.

– Put the chip quickly (without drying) in the IPA for 30 s.

– N2 blowdry.

249


Table E.2: Etch rates (nm/min) of the different layers for different etch gasses.

gas gasflow (sccm) PMMA S1805 W Re SiO2

O2 20 141 63 0 0 -SF6 12.5 164 76 - 20 0

CHF3/O2 50/1.2 51 21 - - 12

Table E.3: Specifications of the different etch steps.

etch gasflow He flow O2 flow pressure power etchtime(sccm) (sccm) (sccm) (μbar) (W) (s)

SF6 (W) 12.5 9.6 0 10 45 25O2 20 0 - 4 50 520

SF6 (Re) 12.5 9.6 0 10 40 70CHF3/O2 50 0 2 8 45 500

Dry etchingWe have used a Leybold Heraeus plasma etch system. It is important that thechamber of this system is clean. Due to preceding etches, a thin contaminationlayer could be present at the walls of the chamber. We found that too muchchamber contamination results in (chip) surface contamination. Therefore it isrecommended to always clean the chamber by mechanical brushing before use.After brushing the chamber has to be cleaned with a 200W oxygen plasma for30 mins.

It is important that the water cooling is always on when the plasma is on. Turnoff the water cooling when the chamber is vented.

In Tab. E.2 we present the etchrates for the different layers in the etchmask.These can be used to modify the etch recipe if this is needed. In Tab. E.3 wesummarize the important etch parameters for the different etch steps. Below wedescribe the process flow in detail.

� SF6 etch W layer:

Preconditioning plasma:

– Open SF6 and He lines. Put the SF6 flow controller to 50 (equals 12.5sccm) and the He flow controller to 5.5 (equals 9.6 sccm).

– Adjust chamber pressure controller until pressure reaches 10μbar.

– Switch on power source and put power to 45 W.

– Optimize bias voltage by adjusting ground and capacitance voltages.Typical values are Vgnd = 557V and Vcap = 327V, resulting in Vbias =−400V.

– Readjust power to 45 W.

– Wait 5 min.

250


– Switch off power source and pressure/flow controllers, switch off vac-uum.

Etch:

– Place chip and tungsten testchip in the center of the chamber in sucha way that contrast on the testchip is visible with the infrared camera.

– Pump down chamber.

– Turn on flow controllers SF6 and He lines. Turn on chamber pressurecontroller.

– Turn on power source. As soon as the plasma will light, the etch willstart. After roughly 10 s all the contrast on the tungsten testchip willbe gone. To be sure that all the tungsten is etched away, etch a bitlonger, 25 s is fine.

– Switch off power source and pressure/flow controllers, switch off vac-uum. Remove chips to prepare plasma for the next etch step.

� S1805 and PMMA etch:

See W etch description. Some etch characteristics:

– O2 flow is 10 (equals 20 sccm).

– Chamber pressure controller is adjusted to reach 10μbar chamber pres-sure.

– Source power is 40 W at Vgnd = 552 V and Vcap = 300 V, resulting inVbias = -360 V.

– In this case, there is no test chip. During the etch, focus at a positionon the chip where contrast is visible due to the tungsten etching. Focuson a part where the tungsten is etched away! Watch the interferencepattern of the reflectometer. During the S1805 and PMMA etch thisshould be a regular peak pattern (roughly 3 peaks), which flattens assoon as the Re layer is reached. The etch time is roughly 8-9 min.

� Re etch:


– SF6 flow is 50 (equals 12.5 sccm) and He flow is 5.5 (equals 9.6 sccm).

– Chamber pressure controller is left at open position, resulting in∼2.5μbarchamber pressure.


– The test chip should be placed such that contrast between Re andSiO2 is visible. After roughly 40 s all the contrast on the Re testchipwill be gone. To be sure that all the Re is etched away, etch a bitlonger, 70 s is fine.

251


� SiO2 etch:


– CHF3 flow is 50 (equals 50 sccm) and O2 flow is 1.0 (equals 2 sccm).

– Chamber pressure controller is left at open position, resulting in apressure of ∼ 8μbar.


– In this case, there is no test chip. During the etch, focus at a positionon the chip where contrast is visible due to preceding etch steps. Focuson a part where the Re is etched away. Watch the interference patternof the reflectometer. During the etch, the pattern should be a regularpeak pattern consisting of roughly 1.5 periods. The etchtime is roughly8-9 min.

Remarks: Due to the presence of fluorene ions and organic compounds,teflon will grow at the walls of the trench. A bit of teflon helps to achievea high anisotropy, but too much teflon growth will result in undesirableeffects like blocking of the trench or preventing good contact from the Refilm to the CNT. Adding a small amount of O2 reduces the teflon growth.The O2 flow is optimized to reduce teflon growth as much as possible, butstill achieve a high anisotropy.

After the etch the mask thickness and mask+trench thickness can be mea-sured with the α-stepper. Typical values are resp. 250 nm and 400 nm,indicating a trench depth of 130 nm + 20nm.

� Lift off etchmask:

– Prepare a beaker with hot acetone (55�C).

– Place the chip vertically in the beaker, switch on sonicator and leaveit for four hours.

– Then leave the chip overnight in cold acetone.

– 30’ cold acetone sonicate.

– Spray acetone over the chip.

– Place the chip in IPA (without drying in between) for 30 s.

– N2 blowdry.

� Inspection:

– First check optically if the etchmask is removed properly.

– Carefully check the quality of the trench etch with SEM imaging.Check for shorts, homogeneity of the trench bottom surface, teflonpresence and degree of anisotropy (view sample under an angle).

Remarks: It is also possible to do an extra SEM check after the second etchstep (by then, the etch mask is etched up to the Re layer). If irregularitiesare visible, stop the etching and start all over with the fabrication of a newetch mask.

252


E.2.2 Catalyst deposition

Catalyst patterning

� Resist spinning:

– Spin PMMA495K anisole 6% at 4000 rpm and bake it for 6 min at175�C.

– Spin PMMA950K anisole 3% at 6000 rpm and bake it for 6 min at175�C.

– Measure mask thickness with α-stepper, thickness should be ∼ 400 nm.

� E-beam patterning:

– Mount chip to sample holder.

– Height alignment of the chip has to be within ±5μm of the referenceheight of the sample holder.

– In-plane alignment should be done by using a line of markers.

– Write down the coordinates of the lower left marker.

– Modify layout file: use the coarse beam (93 nm), use automatic markersearch (enter markersize, position and type which is lowerleft and neg-ative).

– Load sample holder into EBPG and perform the writing job specifiedby the layout file. After finishing, unload the sample holder. The chipis ready for development.

� Resist development:

– Prepare beakers, one with MIBK:IPA = 1:3 and one with IPA.

– Gently move the chip through the MIBK:IPA = 1:3 for 90 s.

– Put the chip quickly (without drying) in the IPA for 30 s.

– N2 blowdry.

Catalyst deposition

The catalyst is deposited from the following solution:

� 30 ml Methanol.

� 30 mg Aluminium oxide particles: Aluminiumoxid C, Degussa.

� 40 mg Fe(NO3)3 9H2O: Ferric nitrate Nonahydrate, Fluka.

� 4.1 mgMoO2(acac)2: Bis(acetylacetonato)dioxomolybdenum (VI), Aldrich.

253


The deposition is done as follows:

� Heat oven to 150�C.

� Sonicate the catalyst (CVD CNT catalyst 4.1 mg Mo, made by Sami Sap-maz) for 30 min.

� Prepare 2 beakers with hot acetone (55�C, au bain Marie).

� Prepare 1 beaker with IPA.

� Prepare a pipette by sonicating it for 5 min in methanol.

� After finishing the catalyst sonication, put a few droplets with the pipetteon the chip. The chip should be covered fully. Let the catalyst dry to theair, it is dry when it is not shiny anymore.

� After drying, bake the chip for 10 min at 150�C.

� Take the chip from the oven and move it for 2 min through the hot acetone.Do this always by moving the chip backwards trough the acetone (to preventcatalyst particles contaminating trenches).

� Put the chip in the second beaker with hot acetone. In between the twobeakers, spray with acetone onto the chip to prevent from drying.

� Put the chip in the IPA for 30 s.

� N2 blowdry.

254


E.2.3 Carbon nanotube growth

The CNT growth process is as follows:

� Open oven, switch it on and set temperature setpoint to 900�C.

� Put sample in the quartz tube (marked QT). By using a metal rod, pushit to the center of the tube.

� Close the tube by gently sliding the inlet and exhaust seals over the tubeas far as possible.

� Place the tube in the oven, in such a way that the sample is placed down-stream close to the temperature sensor.

� Open the main and fine valves of the gas bottles.

� Open all flow meters to max for 2 min to flush all lines.

� Close the H2 and CH4 flow meters fully.

� Adjust the Ar flow meter to 13 (1.5 l/min) and flush 2 min to remove allthe CH4.

� Close the oven, the temperature will go to the target value within 15 min.

� If the temperature is fine, open the H2 flow meter to the maximum value(700 ml/min).

� Close the Ar flow meter fully.

� Open the CH4 flow meter to 4 (520 ml/min), by doing this, CNT growthwill start. Grow for 10 min.

� Close the CH4 flow meter.

� Open the Ar flow meter again to 13 (1.5 l/min), leave H2 at the maximumvalue.

� Open the oven to cool down, this will take like roughly 25 min.

� When the temperature is below 180�C, close all the flow meters and all thegas bottles.

� Remove the tube and take the sample out.

255

Appendix F

Superconducting magnetcoil

Materials

� Multifilament SuperconductingWire, NbTi. supercon-wire.com type: 56S53,#Fil 56, Cu:SC 0.9:1, diameter 0.3mm. critical current (at 4.2K) 3T/125A,5T/100A, 7T/55A, 9T/20A.

� Brass (Cu 0.6Zn 0.4) sheet (0.025mm) for contacting wires to NbTi.

� GE Varnish for fixing magnet coils. Supplier: cmr.uk.com CMR/GEVar-100ml (also known as IMI-7031).

� Non-waxed dental floss for fixing magnet coils.

� Some cylinder templates for winding the coils.

� Brush for applying GE Varnish.

� Masking tape to cover the IVC can before fixing the coils.

Procedure

� Wind a number of coils on different cylinder templates. We used (numberof windings /diameter):

coil 1: 100/44mm 100/38mm 60/31mm 100/26mm.coil 2: 100/44mm 100/38mm 80/31mm 80/26mm.

Make sure the leads to the coils are long enough. Don’t kink the wire, itcan lower Ic.

� Bind each coil with dental floss.

� Wrap the IVC in masking tape to protect it from the coils and GE Varnish.Teflon tape is not sufficient, since the SC wire can cut through the teflontape.

257

F. Superconducting magnet coil

� Fix the coils on the field center to the IVC of the fridge using non-waxeddental floss. We use non-waxed because it is believed that waxed dentalfloss does not soak with GE Varnish, making it brittle at low temperatures.

� When positioned, wrap both coils tightly in dental floss. Use a lot of dentalfloss, the coils should not be able to move at all.

� Take the brush and cover the magnet completely with GE Varnish. Let itdry for at least 12 hours.

� Make leads to the NbTi coil from the brass foil.

� Fold a strip of ∼ 3− 4mm from the foil. Wrap it in masking tape. The foilhas a larger cooling surface compared to a thick wire. Brass has lower heatconduction compared to copper.

Figure F.1: Photograph of two winded coils fixed to an IVC can.

Estimated field

� Outer diameter of fitted coils is 2R = 50mm.

� Total number of windings (coil 1 + coil 2) n = 720.

� Equation for B field of one circular loop: B = μ0I/2R.

� We estimate a field of B/I = μ0n/2R = 18mT/A in the loop center. Atz = 2 cm away from the field center, along the centerline (z-axis) of the

loop, we estimate: Bz/A = (μ0/(4π)) 2πR2/

(z2 +R2

)3/2= 8mT/A.

258

Summary

Josephson effects in carbon nanotubemechanical resonators

and graphene

Carbon nanotubes (CNTs) and graphene are novel materials with special prop-erties. We use their low-dimensionality to experimentally study the Josephsoneffect in new regimes, and use the Josephson effect to probe GHz mechanicalvibrations of unprecedented high-Q CNT resonators.

In graphene we have tried to make a π-junction in a magnetic field. Our mainaccomplishment here is the development of graphene Josephson junctions (JJs)that enable supercurrent up to unprecedented high magnetic fields. In one devicewe find a field dependence of the switching current that is expected for a π-junction, but is disputable, likely due to trapped flux.

Our main achievement is the development of clean suspended CNT JJs. We havestudied the AC Josephson effect on mechanical resonance for the first time inCNTs.

Our main finding is a DC current shift on mechanical resonance, that we con-tribute to CNT vibrations mixing with AC Josephson currents. The presenceof superconductivity increases the mixing current by two orders of magnitude,allowing the detection of vibrations as small as the thermal motion.

This mixing enables us to probe the current phase relation in a regime whereDC supercurrent is suppressed. A change of the sign of the mechanical currentbeyond a threshold magnetic field, indicates a Zeeman π-junction in our CNTs.We present a simple theoretical model to account for our findings.

We have studied mechanical current on Shapiro plateaus, and find a non-trivialpower dependence that is inconsistent with regular driving and mixing. We havealso studied mechanical resonance at a fractional driving frequency. Our data isconsistent with parametric excitation of mechanical vibrations, and detection bymixing with Josephson harmonics.

We suggest a broad range of experiments in which the special properties of CNTJJs can be used, and studied, in superconducting devices. We suggest develop-ment of high-field compatible CNT-based Josephson parametric amplifiers andtransmon qubits.

C.J.H. KeijzersLeiden, September 2012.

259

Samenvatting

Josephson effecten in koolstofnanobuismechanische resonatoren

en graphene

Koolstofnanobuizen (KNBs) en grafeen zijn nieuwe materialen met speciale eigen-schappen. We hebben hun lage dimensionaliteit gebruikt om het Josephson effectte bestuderen in een nieuw regime, en gebruiken het Josephson effect om GHzmechanische trillingen te meten van KNBs met een zeer hoge Q-factor.

In grafeen hebben we geprobeerd om een π-junctie te maken in een magneetveld.Onze belangrijkste verrichting is het ontwikkelen van grafeen Josephson juncties(JJs) waarin superstroom is bij ongekend hoge magneetvelden. In een systeemvinden we een magneetveld afhankelijkheid van de schakelstroom die verwachtwordt voor een π-junctie, maar deze is aanvechtbaar, vermoedelijk door vastzit-tend flux.

Onze belangrijkste prestatie is de ontwikkeling van schone en hangende KNB JJs.We hebben voor het eerst het AC Josephson effect bestudeerd op mechanischeresonantie in KNBs.

Onze voornaamste bevinding is een extra gelijkstroom op mechanische resonantie,die we toeschrijven aan KNB trillingen die mengen met AC Josephson stromen.De aanwezigheid van supergeleiding vergroot de mengstroom met twee ordesvan grootte, waardoor het detecteren van trillingen zo klein als de thermischefluctuaties mogelijk is.

Door deze menging zijn we in staat om de stroom-fase relatie te onderzoekenin een gebied waar de DC superstroom is onderdrukt. Een wisseling van hetteken van de mechanische stroom boven een bepaald magneetveld wijst op eenZeeman π-junctie in onze KNBs. We presenteren een eenvoudig theoretisch modelwaarmee we onze bevindingen verklaren.

We hebben mechanische stroom onderzocht op Shapiro plateaus, en we vinden eenniet-triviale vermogen afhankelijkheid die inconsistent is met normale excitatie endetectie. We hebben ook de mechanische resonantie onderzocht bij een fractioneleexcitatie frequentie. Onze data is consistent met parametrische excitatie vanmechanische vibraties en detectie door menging met Josephson harmonischen.

We stellen een brede reeks van experimenten voor, waarin de speciale eigen-schappen van KNB JJs gebruikt –en bestudeerd– kunnen worden in supergelei-dende systemen. We stellen voor om hoog-magneetveld compatibele op KNBgebaseerde, Josephson parametrische versterkers en transmon qubits te ontwikke-len.

C.J.H. KeijzersLeiden, september 2012.

261

Acknowledgements

I color the sky with you,I let you choose the blue.Everyone needs an editor - Mates of State

Above most, I want to thank all my (past and present) colleagues in the QuantumTransport group for all the good times, help and support, in the last six years.Without you, this work would not have been possible, and my time in QT wouldhave been much less exciting.

I want to start by thanking my main collaborators. Dear Leo, I’m very grateful toyou for giving me the opportunity to do research in your group. It has been a veryinteresting time. With your passion and confidence, you naturally motivate yourstudents and colleagues to excel. For me this is what makes you an outstandingleader. Thank you for all your help and support.

Dear Sergey and Gary, thank you very much for all your help and support. Iwish you both the best in your future careers. Dear Vincent and Ciprian, youhave contributed much to this work and I’m very thankful for your help. Vincent,it was a pleasure to work with you during your master thesis project. I’m veryhappy to see that you enjoy your time in QT. Dear Ciprian, thank you verymuch for our countless discussions. Your continuous optimism and support wasimportant to me. Dear Yuli, thank you for your interest, advice and support.

In particular I thank Atilla, Edward, Leo DiCarlo, Sergey and Stijn for providingfeedback while I was writing my thesis. I’m grateful to Martin Weides for pro-viding rhenium films from UCSB in the early stages of the CNT project. Thanksto Oleh Klochan and Tungky Subroto for providing graphene when I startedthe graphene project. Thanks to Masafumi Jo for his help during my first yearnanowire project. I was in the fortunate position to be advised by many out-standing people. I want to thank Jan Aarts, Leo DiCarlo, Cees Harmans, TeunKlapwijk, Konrad Lehnert, Hans Mooij, Alberto Morpurgo, Raymond Schouten,Lieven Vandersypen and Ad Verbruggen. Our discussions (with some of youmany, and with others few) have all contributed in one way or another to thiswork.

There are a few people I want to especially thank for the good times we hadtogether. Tristan, thank you for being my mentor and guide when I arrived inQT, those were the days. Dear Marc, thank you for helping me when I started

263

Acknowledgements

my work in the nanowire team. You’re also a good friend and it was always funto listen to your interesting stories and it was great to spend time together inBeijing. In my first year I worked together with Stevan and Juriaan. Juriaan,your optimism and motivation were also encouraging to me. Stevan thank youvery much for being a great friend and for all the discussions we had.

Stijn, you have a natural tendency to help others, something I really admire. DearToeno, I appreciate our discussions and the party-activities we had together verymuch, and I’m looking forward to continue them in the future. Dear Victor, itis a pleasure to know you for such a long time and it was great to have you asan office-mate. I hope we’ll have more chats in the future. My other office mateswere Georg, Kun and Erika. Georg it was nice to face you for so many years!Kun you are a really nice guy and you combine a very good sense of humor withan eye for detail. I’ll miss your daily company. Thanks for your interest in, andplay times with Tex. Erika you’ve surely increased the temperature gradient inour room!

Arjan, Fei and Tim, good luck with your projects, I have enjoyed our conver-sations very much. Maria, you are a great colleague and also a good QT trip-organizer. Thanks for your companionship and the nice times we had together.Amelia, thank you for the good times we had together, I’ve enjoyed them verymuch. Katja, thanks for your companionship in Beijing, it was really fun to dotourist activities together. The same holds for Alina, whom I also have to thankfor the fun swimming activities at the countless FOM conferences we have at-tended. I’m also thankful to Pieter for the nice discussions we had during ourmeasurement time in neighboring labs, and to Vlad for lending me his underwa-ter goggles. Reinier, thank you for our conversations. It is very impressive tome how you’ve managed your research so well. Nika and Michael, thanks for themany nice conversations we had. I consider your attitude towards work and lifeas a good example, and I’m very happy to have met you. Michael, Ciprian andStevan, partying with you in Tokyo was one of the highlights in the past six yearsfor me.

Much of my work would not have been possible if Diego had not helped me outso many times (for example by on-demand switching measurement modules evenpast midnight). Remote desktop would have been useless without his help! Thesame holds for Vincent and Kun, which I have to thank again for no-problemtransferring helium many times, whenever I could not do it myself.

Thanks to Bram, Remco, Peter and Jelle for maintaining a seemingly infinitesupply of helium and other utilities and services. It was fun to repair the fridgetogether with you Jelle, I hope you learned something! Whenever I needed tohave something machined in the workshop, Aad was there and helped me out,thank you Aad. It was amazing to see how you made one bolt consisting out oftwo bolts, to remove half a bolt. In the cleanrooms I was supported by manyexperienced and helpful people. In particular I want to thank: Anja, Arnold,Emile, Ewan, Hozanna, Marc, Marco 1, Marco 2, Patrick, and Roel. I also haveto thank our secretary, Angele, Dominique, Marja and Yuki. You were not onlyalways very helpful in arranging important stuff for me, but also by having casualchit-chat every now and then.

264

Acknowledgements

In the past six years most people that were around when I started my work inQT have left the group, I’ve seen over a hundred people entering and leaving QT.Now it is also time for me to leave, and I’m very happy that I can do this nowmy work is finished, but not before I thank my family.

Pap en mam, dankjewel voor al jullie hulp en ondersteuning op de lange wegdie nu uiteindelijk tot dit proefschrift heeft geleid. Pap, het meeste heb ik vanjou geleerd en daar ben ik je heel dankbaar voor. Margo en Richard, Hans enAna-Iris, Marieke en Paul, Carin en Noach, dankjewel voor jullie steun!

Het meeste dank ben ik verschuldigd aan mijn lieve vriendin Ionica. Lieve Ionica,dankjewel voor alles, en voor Tex in het bijzonder.

Han KeijzersLeiden, September 2012

265

Curriculum Vitae

Christianus Johannes Henricus (Han) KEIJZERS

Sep. 3, 1980 Born in Deurne, The Netherlands.

1992 - 1996 Junior general education, Bernard Alfrink Mavo, Deurne

1996 - 2000 Senior vocational education, Ter AA College, HelmondElectronics

2000 - 2004 B.App.Sc. Applied Physics,Fontys School of Higher Professional Education, EindhovenMedical EngineeringThesis work at the department of anesthesiology atErasmus MC, Rotterdam, on bioimpedance spectroscopyfor patients with chronic pain.

2004 Junior researcher, Erasmus MC, Rotterdam (six months)Department of anesthesiology

2004 - 2007 MSc. Applied Physics, Delft University of TechnologyQuantum Transport groupThesis work in the group of Prof.dr.ir. L.M.K. Vandersypenon spin readout in quantum dots.

2007 - 2012 PhD candidate, Applied Physics, Delft University of TechnologyQuantum Transport groupThesis: Josephson effects in carbon nanotube mechanicalresonators and graphenePromotor: prof.dr.ir. L.P. Kouwenhoven.

267

Documents

Josephson effects in carbon nanotube mechanical resonators and