Nanomechanical shuttle transfer of electrons

R. I. Shekhter∗,1 L. Y. Gorelik,2 M. Jonson,1 Y. M. Galperin,3, 4, 5 and V. M. Vinokur51Department of Physics, Goteborg University, SE-412 96 Goteborg, Sweden†

2Department of Applied Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden3Department of Physics, University of Oslo, PO Box 1048 Blindern, 0316 Oslo, Norway

4A. F. Ioffe Physico-Technical Institute of Russian Academy of Sciences, 194021 St. Petersburg, Russia5Argonne National Laboratory, 9700 S. Cass av., Argonne, IL 60439, USA

(Dated: December 20, 2004)

The coupling between mechanical deformations and electronic charge transport in nanostructuresand in composite materials with nanoscale components gives rise to a new class of phenomena —nanoelectromechanical transport — and opens up a new route in nanotechnology. The interplaybetween the electronic and mechanical degrees of freedom is especially important in nanocompositesconsisting of materials with very different elastic properties. Mechanical degrees of freedom takeon a primary role in the charge transfer process in many single-electron devices, where transport iscontrolled by quantum-mechanical tunnelling and Coulomb interactions, but where tunnel barrierscan be modified as a result of mechanical motion. A typical system of this kind is a single-electrontransistor (SET) with deformable tunnel barriers, a so called Nano-Electro-Mechanical SET (NEM-SET). The new kind of electron transport in this and other types of nanodevices is referred to as“shuttle transport” of electrons, which implies that electrons is transferred between metallic leadsvia a movable small-sized cluster. The present review is devoted to the fundamental aspects ofshuttle transport and to a description of major developments in the theoretical and experimentalresearch in the field. Prospective applications of this exciting phenomenon that remarkably combinestraditional mechanics of materials with the most advanced effects of quantum physics, will also betouched upon.

PACS numbers: 68.60.Bs,73.23.-b,87.80.Mj

Contents

1. Introduction 2

2. Single-electron transfer by a nanoshuttle 42.1. Classical shuttling of particles 5

2.1.1. Requirements for incoherent transport 52.1.2. Shuttling of electrical charge by a

movable Coulomb dot 52.1.3. Shuttling in dissipative nanostructures 72.1.4. Accuracy of a mechanical single-electron

shuttle. 82.1.5. Gate voltage control of shuttle

mechanics 92.1.6. Nanoparticle chains 102.1.7. Charge shuttling by a suspended

nanotube 102.2. The charge shuttle as a nanomechanical ratchet112.3. Classical shuttling of electron waves 122.4. Charge transfer through a quantum oscillator 132.5. Spin-dependent transport of electrons in a

shuttle structure 152.5.1. Nanomechanical manipulation of

nanomagnets 152.5.2. Spintronics of a nanoelectromechanical

shuttle 17

∗Corresponding author†Electronic address: shekhter@fy.chalmers.se

3. Experiments on electron shuttling 18

4. Coherent transfer of Cooper pairs by amovable grain 214.1. Requirements for shuttling of Cooper pairs 214.2. Parity effect and the Single-Cooper-pair box 224.3. Basic principles 22

4.3.1. Scattering and free motion 234.3.2. Hamiltonian 24

4.4. Transferring Cooper pairs between coupledleads 24

4.5. Shuttling Cooper pairs between disconnectedleads 25

5. Noise in shuttle transport 275.1. General concepts 275.2. Incoherent electron transport and classical

mechanical motion 285.3. Noise in a quantum shuttle 305.4. Driven charge shuttle 315.5. Noise in Cooper pair shuttling 32

6. Discussion and conclusion 34

Acknowledgments 34

References 34

2

1. INTRODUCTION

In 1911 Robert Millikan of the University of Chicagopublished the details of an experiment that proved be-yond doubt that charge was carried by discrete positiveand negative entities of equal magnitude, which he calledelectrons [1]. The discovery that charge is quantized wasimmediately recognized as very important and he was ac-cordingly awarded the Nobel Prize for Physics in 1923.But although it took the scientific community almost sev-enty years to appreciate it, Millikan’s oil-drop experimentalso marked the beginning of what we now call singleelectronics. To see why, it is important to focus on twoessential features of Millikan’s set-up, which turn out tobe important principles of single-electronics. The first isthat he was able to add charge to small pieces of mat-ter, in his case oil-droplets with radii of about 1 µm, thesecond is that he studied the mechanical motion of thesedroplets in an external electric field in order to determinetheir charge. Both these elements of his experiment even-tually became important principles for the operation ofsingle-electron devices and critical for their applications.However, an important condition for these principles tobe relevant in a device had first to be fulfilled – one hadto learn how to accurately control both the charging pro-cess and the mechanics of the “oil-drops”. To achievethis at the submicron level required almost seventy yearsof technological progress.

A first step towards modern single electronics wastaken in the late 1980’s by two groups in Russia and theUSA [2]. Using different techniques they were both ableto make well-characterized metallic islands of submicronsize and charge them in a controllable way. Today, ex-periments have become even more sophisticated and thedegree of control is much higher.

The ongoing miniaturization of electronic devices leadsto an enhanced role for electrical charging and, as a re-sult, to significantly larger mechanical forces between dif-ferent elements of the device. Consequently, mechanicaldeformations are produced, which in turn influence thedistribution and transfer of charge. Since the electricaland mechanical energy scales are comparable the aboveinterplay results in coupled electro-mechanical modes. Inthis way the original Millikan setup is recovered at a newnanoscopic level.

Electromechanical interactions are important not onlyin the devices for nanoelectronics. They are significantfor charge transport through a variety of conducting me-dia containing nanoscale components. For example, ifelectrons can be trapped in small conducting regionswithin an insulating matrix, quantum-mechanical tun-nelling and Coulomb interactions turn out to be the mainmechanisms controlling electron transport. In particu-lar, this can be the case in granular materials, where theCoulomb blockade phenomenon reduces charge fluctua-tions and leads to single-electronic tunnelling.

Mechanical deformability plays a special role in nano-materials. On the one hand, it originates from the elec-

tronic contribution to the elastic constants of the mate-rial and, on the other hand, it has tremendous impacton the electron tunnelling. The interplay between elec-tronic and mechanical degrees of freedom is especiallyimportant in nanocomposites consisting of materials withvery different elastic properties. Liquid nanocompositesor electrolytes represent an extreme case of such systems.There the charges are localized on ions, and it is the me-chanical, convective motion of these “nanocarriers” thatis responsible for the charge transfer (the ionic transportmechanism).

In solid-state composite materials the higher densityof carriers together with a significant freezing of theirmechanical motion increases the importance of inter-particle electron tunnelling and makes this charge trans-port mechanism competitive with mechanical convection.Hence we arrive at the very interesting situation whenelectrical and mechanical degrees of freedom cannot beseparated. One has to consider a new — nanoelectrome-chanical — type of transport.

This is the situation occurring in many metal-organicnanocomposites, where small conducting nanoparticlesor “dots” are embedded in a soft organic molecular ma-trix. There, on-dot discrete charge fluctuations causedby electrostatically induced mechanical distortions playan essential role providing a feedback to the mechani-cal motion. Both the response to external perturbationsand the noise properties of such materials are qualita-tively different from those known for bulk homogeneousconductors.

During the last decade or so, nanotechnology has ad-vanced the ability to fabricate systems in which chem-ical self-assembly defines the functional and structuralunits of nanoelectronic devices [3]. Since the elastic pa-rameters of many compounds and devices currently uti-lized can be much “softer” than those of semiconductorsand metals, mechanical degrees of freedom may play animportant role in charge transfer. In particular, chargetransfer via tunnelling through a device can be dramat-ically enhanced by the mechanical motion of some partof the device.

Recently, nanomechanical oscillators [4] have beencombined with single-electron tunnelling [5] devices, re-sulting in a new class of nanoelectromechanical sys-tems (NEMS). Experiments measuring electron trans-port through single oscillating molecules [6–9], suspendedsemiconductor systems [10, 11], and suspended carbonnanotubes [12] clearly demonstrate the influence of me-chanical degrees of freedom on the current in the single-electron tunnelling regime.

As a simple example of a device of this type, consider ametallic grain elastically suspended between a source anda drain electrode as in Fig. 1. Since the central conduct-ing grain can move, we will refer to this device as a nano-electromechanical single-electron transistor (NEM-SET)rather than just a single-electron transistor (SET). If, dueto a fluctuation in its position, the grain were to comeclose to the source (or drain) electrode the tunnelling cou-

3

V/2 V/2

a)

RL RRm

CLCR

"loading" of 2N electrons

"unloading" of 2N electrons

V/2V/2

b)

q = ne

q = −ne

E = αV

FIG. 1: (a) Simple model of a soft Coulomb blockade systemin which a metallic grain (center) is linked to two electrodesby elastically deformable organic molecular links. (b) A dy-namic instability occurs since in the presence of a sufficientlylarge bias voltage V the grain is accelerated by the corre-sponding electrostatic force towards first one, then the otherelectrode. A cyclic change in direction is caused by the re-peated “loading” of electrons near the negatively biased elec-trode and the subsequent “unloading” of the same at the pos-itively biased electrode. As a result the sign of the net graincharge alternates leading to an oscillatory grain motion anda novel “electron shuttle” mechanism for charge transport.From Ref. [13], L. Y. Gorelik et al., Phys. Rev. Lett. 80, 4526(1998), with permission from the American Physical Societyc© 1998.

pling between them would increase significantly and thegrain would be negatively (positively) charged. Then, ac-celerated by elastic and Coulomb electrostatic forces, thegrain would move back and approach the drain (source)electrode, thus transferring the acquired charge. The de-scribed process is usually referred to as ”shuttling” ofelectrons. In general, the shuttle mechanism can be de-fined as a charge transfer through a mechanical subsys-tem facilitated by its oscillatory center-of-mass motion.The key factor here is that in shuttling the charge ofthe grain, q(t), is correlated with the grain velocity, x(t),in such a way that the time average q(t)x(t) 6= 0. Itfollows that the average work performed by the electro-static force is nonzero even if x(t) = 0, and as a resultthe mechanical motion and charge transfer can be un-stable with respect to the formation of periodic or quasiperiodic mechanical motion and electrical signal.

Being induced by the coupling between tunnelling elec-trons and vibrational degrees of freedom, shuttle trans-port should be discriminated from the conventionally dis-cussed vibron-assisted inelastic tunnelling. The differ-ence is that shuttling results from an electromechanical

instability. If the mechanical motion is strongly dampedthe device is mechanically stable, and the number of gen-erated vibrons is close to the equilibrium value. In thissituation the notion of vibron-assisted tunnelling is ad-equate. As the damping in the mechanical system de-creases, or the driving voltage increases one may reach apoint where the mechanical stability of the device is lost.

At this point the number of generated vibrons increasesand reaches a large value. Moreover, a special type of co-herence in the mechanical system is maintained due tothe coupling to tunnelling electrons. This coherence canbe characterized by nonzero off-diagonal elements of thedensity matrix in the vibron number representation. Asa result the vibrational degrees of freedom can be de-scribed by a classical field representing the mechanicaldisplacement of the shuttle. In this – shuttle – regimethe electron-vibron interaction develops into a mechani-cal transportation of electrons.

In the following we will focus on the shuttling transportof charge. A large amount of both theoretical [14–29]and experimental [30–32] papers concerning the electron-vibron interaction in single-molecule SET devices haveappeared in the literature. However, since they are notrelated to the shuttle instability and shuttling of charge,which is the subject of the present review, we will leavethem out and refer the interested reader to other recentreviews of more general issues in nanomechanics [4, 33–35].

An important feature of nanosystems is the Coulombblockade [36] phenomenon. A small, initially neutral sys-tem that has accepted an extra electron, becomes nega-tively charged, and under certain conditions another elec-tron cannot, due to Coulomb repulsion, reach the grain.As a result, it has to wait until the first electron has es-caped. Until then further transport is blocked. Thus,under Coulomb blockade conditions the electrons can betransferred only one by one (or, more generally, in inte-ger numbers). The smaller the system capacitance, thebigger is the charging energy. Consequently, Coulombblockade is an intrinsic property of small devices, and itsimportance increases with the progress in nanoscienceand nanotechnology. Hence, due to Coulomb blockade,shuttling of single electrons or single Cooper pairs cantake place.

During the past seven years, shuttles of different typeshave been studied theoretically and experimentally. Wehave just emphasized that shuttling as opposed to vibron-assisted tunnelling is due to an intrinsic instability. How-ever, it can also be due to an externally driven grain mo-tion using, e.g., an AC electric or mechanical force. An-other point of interest is that electron transport througha NEM-SET device can occur in regimes where either aclassical or a quantum mechanical theory is called for.The mechanical degrees of freedom may also require ei-ther a classical or a quantum-mechanical description.Hence, shuttle charge transfer in soft nanostructures in-volves rich and interesting physics.

Presently, many researchers are interested in shuttling

4

in nanoelectromechanical systems aiming to determinefundamental properties of electro-mechanical coupling innanostructures, electron and phonon transport, etc. Thisknowledge will definitely give renewed impetus to the de-sign of new applications such as, e.g., nanogenerators,nanoswitches and current standards.

Shuttle electron transfer can take place not onlythrough a metallic grain, but also through a relativelysoft nanocluster. In this connection, it is importantto study vibrational modes of electromechanical systemswith several degrees of freedom. If the center-of-massmotion can be clearly separated from other modes, thenthe shuttle electron transfer though the system is similarto that in the case of a rigid grain. Otherwise, it is moreappropriate to speak of vibron-assisted tunnelling ratherthan shuttling.

Further, electromechanical coupling and shuttle trans-port is not only a feature of heteroelastic nanocompos-ites. It also has, as will be seen below, relevance for othernanoelectromechanical systems intentionally designed towork at the nanometer scale [37–39]. Early work on shut-tle transport of electrons were reviewed in Ref. [40].

Our aim is here to discuss the issues mentioned abovein more detail. We will start from the simplest case ofa particle with one mechanical degree of freedom locatedbetween two electrodes and elastically coupled either tothe substrate or to the leads. This situation is relevant toseveral experiments, e. g., to electron shuttling throughan artificial structure [41], and to shuttling through anoscillating C60 molecule [6]. Both classical and quantumelectron transport will be considered in Sections 2.1 and2.3, respectively (see Table I). It will be shown that inboth cases an electromechanical instability occurs whichleads to a periodic mechanical motion of the elasticallysuspended particle [13, 42, 43]. In Sec. 2.4 our aim is toreview theoretical work regarding systems where the me-chanical degrees of freedom need to be treated quantummechanically. A review of the available experimental re-sults is given in Sec. 3. Finally, the possibility of coherentshuttling of Cooper pairs between two superconductorsthrough a movable superconducting grain will be con-sidered in Sec. 4, and noise in different types of shuttlesystems in Sec. 5.

2. SINGLE-ELECTRON TRANSFER BY ANANOSHUTTLE

As already mentioned, shuttle charge transfer involvestwo distinct charge transfer processes: the tunnelling ofcharge between the leads and a (moving) cluster as wellas the mechanical, “convective” motion of the chargedcluster. The mechanical motion of the shuttle can obeyeither classical or quantum mechanics. A classical de-scription is sufficient if the force that acts on the shuttledepends only weakly on its spatial position. In generalthis requirement can be formulated as

dλ(x)/dx ¿ 1 , (1)

Mechanical motion

Ele

ctr

on

tra

nsp

ort

classical quantum

incohere

nt

cohere

nt

Classical shuttling of

particles

Sec. 2.1

Quantum shuttling of

particles

Sec. 2.3

Classical shuttling of

waves

Sec. 2.2, Sec. 4

Quantum shuttling of

waves

Sec. 2.3

TABLE I: Classification of shuttle transport. Shuttle trans-port of charge can be categorized according to what type ofphysical description is needed for the mechanical and elec-tronic subsystems.

where λ(x) = ~/p(x) is the effective de Broglie wave-length of the shuttle and p(x) is its classical momentumat a position x on its classical trajectory.

Now, even if the criterion (1) for the shuttle motion tobe classical is fulfilled, the tunnelling of electrons throughthe grain can be either sequential or coherent. In the firstcase, the relevant physical picture is fully classical — theelectrons can be regarded as classical particles and theirtransport properties can be described by a Master equa-tion. We will refer to this situation as classical shuttlingof particles. In the case of coherent tunnelling, the elec-trons should be treated as wave packets while keepingtrack of the properties of their wave functions. In thissituation we will talk about classical shuttling of waves.The word “classical” here serves as a reminder that themechanical motion is still classical. All recent experi-ments can be interpreted as classical shuttling satisfyingthe condition (1). If the dissipation is low enough, see thediscussion in Sec. 2.4, this is true also for the experimentof Park et al. [6] involving a C60 molecule. However, inprinciple, the condition (1) may not be fulfilled. If so,the mechanical motion has to be described by quantummechanics. We refer to this case as quantum shuttling ofeither particles or waves, depending on the nature of theelectron tunnelling process. In the following Sections allfour regimes will be considered (see Table I). Before weproceed, however, a small digression about the condition(1) is in order.

Whether the criterion (1) for the shuttle motion to beclassical is fulfilled or not depends on how quickly theshuttle momentum p(x) varies with position x. This inturn is determined by the forces acting upon the clus-ter. The character of the shuttle transport is somewhatcomplicated by the fact that the electron is not necessar-ily localized at the moving shuttle cluster, it can also be

5

extended between the cluster and the leads as a resultof quantum delocalization. Since the forces that act onthe cluster are different in the different situations, theconcrete forms of the criterion (1) can also be different.If the electron is localized at the cluster, the forces aredue to the direct Coulomb interaction between the leadsand the excess cluster charge, and the criterion (1) canbe cast into the form x0 ¿ λ. Here x0 =

√~/mω0 is the

quantum-mechanical zero-point vibration amplitude, ω0

is the angular eigenfrequency of cluster vibrations, whileλ — to be defined in Sec. 2.1.2 below — is the charac-teristic (decay) length of the tunnelling coupling. Evenif this condition is not met, many properties of shuttletransport can be understood classically provided the os-cillation amplitude, A, is so large that A À x0. In thissituation the region where quantum effects are importantis relatively narrow compared to the whole trajectory.

Quantum delocalization of electron states betweencluster and leads gives rise to what one may call cohe-sive forces and makes the situation more complicated.In Sec. 2.4 we will show that for sufficiently small biasvoltages (and hence for sufficiently weak electric fields) aspecific quantum regime appears, where even relativelylarge cluster vibration amplitudes, A & λ À x0, do notallow a classical description of the shuttle transport.

2.1. Classical shuttling of particles

We begin this Section by considering the classical shut-tling of electrons and by specifying the conditions whichhave to be met for shuttling to fall into this category(Sec. 2.1.1). We then proceed to considering the classicalshuttling of electrons by a harmonically bound clusterbetween two leads in Sec. 2.1.2. After that, having seenthat low damping is necessary for shuttle transport in thiscase, we turn to studying a system dominated by viscousforces in 2.1.3. We will find that classical shuttling ofparticles can take place also in this case.

The prospects for finding applications of the shuttletransport mechanism strongly depend on resolving sev-eral issues. Among them are: (i) what are the conditionsfor ideal shuttling — crucially important for applicationssuch as standards of electric current or for sensors; (ii) isit possible to achieve gate-controlled shuttling, which isimportant for single-electron transistor applications; (iii)what is the role of other mechanical degrees of freedomthan those associated with the center-of-mass motion.These issues will be discussed later in this Section, insubsections 2.1.4, 2.1.5 and 2.1.6.

2.1.1. Requirements for incoherent transport

A schematic picture of a single-electron tunnelling de-vice with a movable metallic cluster as its central ele-ment is presented in Fig. 1. In this case center-of-massmechanical vibrations of the grain are allowed (consider

the elastic springs that connect the central electrode tothe leads in Fig. 1). Since electronic transport throughthe device requires electrons to tunnel between the leadsand the central small-size conducting grain, it is stronglyaffected by any vibration-induced displacements of thegrain.

A number of characteristic times determine the dy-namical evolution of the system. Electronic degrees offreedom are represented by frequencies corresponding tothe Fermi energies in each of the conductors and to theapplied voltage V . In addition one has an inverse relax-ation time, an inverse phase breaking time for electrons inthe conductors and a charge relaxation time, ω−1

R = RC,due to tunnelling. Here R and C are the resistance andcapacitance of the tunnel junction, respectively. Mechan-ical degrees of freedom are characterized by a vibrationfrequency ω0. The condition that ~ωR should be muchsmaller than the Fermi energy is the standard conditionfor a weak tunnelling coupling and holds very well in typ-ical tunnel structures. Since a finite voltage is supposedto be applied, causing a non-equilibrium evolution of thesystem, the question of how fast is the electronic relax-ation becomes relevant. Two possible scenarios for thetransfer of electrons through the metallic cluster can beidentified depending on the ratio between the tunnellingrelaxation time ω−1

R and the relaxation time τ0 of elec-trons on the grain. In the case where τ0 is much shorterthan ω−1

R , the two sequential tunnelling events that arenecessary to transfer an electron from one lead to theother through the grain cannot be considered to be aquantum mechanically coherent process. This is becauserelaxation and phase breaking processes occur in betweenthese events, which are separated by a time delay of or-der ω−1

R . On the contrary, all tunnelling events betweeneither lead and the grain are incoherent, i.e., independentevents. Fast relaxation of electrons in all three conduc-tors is supposed to be responsible for the formation ofa local equilibrium distribution of electrons in each con-ductor. This is the approach, which we will use in thepresent Section. In the opposite limit, i.e. when τ0 ismuch larger than ω−1

R , quantum coherence plays a dom-inating role in the electronic charge transfer process andall relaxation takes place in the leads far away from thecentral part of the device. This case will be consideredin Section 2.3.

2.1.2. Shuttling of electrical charge by a movable Coulombdot

The tunnel junctions between the leads and the grainin Fig. 1 are modelled by tunnelling resistances RL(x)and RR(x) which are assumed to be exponential func-tions of the grain coordinate x. In order to avoid unim-portant technical complications we study the symmetriccase for which RL,R = R(0)e±x/λ, and where we will re-fer to λ as the tunnelling length. When the position ofthe grain is fixed, the electrical potential of the grain and

6

its charge qst follow from balancing the current betweenthe grain and the leads [36]. As a consequence, at agiven bias voltage V the charge qst(x) is completely con-trolled by the ratio RL(x)/RR(x) and dqst(x)/dx < 0. Inaddition the bias voltage generates an electrostatic fieldE = αV in the space between the leads and, hence, acharged grain will be subjected to an electrostatic forceFq = αV q.

The central point of our considerations is that thegrain — because of the “softness” of the links connect-ing it to the leads — may move and change its position.The grain motion disturbs the current balance and asa result the grain charge will vary in time in tact withthe grain displacement. This variation affects the workW = αV

∫xq(t)dt performed on the grain during, say,

one period of its oscillatory motion.It is significant that the work is nonzero and positive,

i.e., the electrostatic force, on the average, acceleratesthe grain. The nature of this acceleration is best un-derstood by considering a grain oscillating with a fre-quency which is much lower than the typical charge fluc-tuation frequency ωR = 1/RC. Here C is the capaci-tance of the metallic cluster which for room temperatureCoulomb-blockade systems is of the order 10−18 − 10−19

F. In this limit the charge deviation δq ≡ q− qst(x) con-nected with retardation effects is given by the expressionδq = −ω−1

R xdqst(x)/dx. Hence the extra charge δq de-pends on the value and direction of the grain velocityand as a consequence, the grain acts as a shuttle thatcarries positive extra charge on its way from the positiveto the negative electrode and negative extra charge on itsreturn trip. The electrostatic force δFq = αV δq is thusat all times directed along the line of motion causing thegrain to accelerate. To be more precise, it has been shown[13, 42] that for small deviations from equilibrium (x = 0,q = 0) and provided q(t) is defined as the linear responseto the grain displacement, q(t) =

∫χ(t − t′)x(t′)dt′, the

work done on the grain is positive for any relation be-tween the charge fluctuation frequency ωR and the fre-quency of the grain vibration.

In any real system a certain amount Q of energy isdissipated due to viscous damping, which always exist.In order to get to the self-excitation regime, more energymust be pumped into the system from the electrostaticfield than can be dissipated; W must exceed Q. Sincethe electrostatic force increases with the bias voltage thiscondition can be fulfilled if V exceeds some critical valueVc.

If the electrostatic and damping forces are muchsmaller than the elastic restoring force, self-excitation ofvibrations with a frequency equal to the eigenfrequencyof the elastic oscillations arise. In this case Vc can be im-plicitly defined by the relation ω0γ = αVcImχ(ω), whereω0γ is the imaginary part of the complex dynamic mod-ulus. In the general case, when the charge response isdetermined by Coulomb-blockade phenomena, χ is an in-creasing but rather complicated function of V and thereis no way to solve for Vc analytically. However, one can

show [42] that the minimal value of Vc corresponds tothe situation when the charge exchange frequency ωR isof the same order as the eigenfrequency ω0 of the grainvibrations.

Above the threshold voltage, the oscillation amplitudewill increase exponentially until a balance between dis-sipated and absorbed energy is achieved and the systemreaches a stable self-oscillating regime. The amplitudeA of the self-oscillations will therefore be determined bythe criterion W (A) = Q(A).

The transition from the static regime to the self-oscillating can be associated with either soft or hard ex-citation of self-oscillations depending on the relation be-tween the charge exchange frequency ωR and the grainoscillation eigenfrequency ω0 [42]. Soft excitation takesplace if ωR/ω0 > 2

√3. In this case the amplitude of

the stable self-oscillation regime increases smoothly (withvoltage increase) from zero at the transition voltage. Ina case of hard excitation (ωR/ω0 < 2

√3) the oscillation

amplitude jumps to a finite value when voltage exceedsVc. It was also found [42] that the hard excitation is ac-companied by a hysteretic behavior of the current-voltagecharacteristics.

An important simplification in early theories of shut-tling [42] was that the nanomechanical coupling was sup-posed to be weak. In other words the Coulomb forceacting on the grain was assumed to be small comparedwith the elastic force responsible for the mechanical vi-brations. An important extension of the theory was thenmade in Refs. [44] and [45], where the full semiclassi-cal analysis was extended to the case of an asymmetricshuttle device and arbitrarily strong nanoelectromechan-ical coupling strength. This becomes possible by usinga local-in-time master equation approach. The time de-pendent master equation was solved numerically in theregimes of both semiclassical and quantum dynamics.It follows from both a quantum and a classical analy-sis [44, 45] that the shuttle current depends linearly onthe equilibrium position of the vibrating dot. This posi-tion may be shifted by an external magnetic force actingon a spin-polarized dot if an inhomogenous magnetic fieldis applied. A theoretical analysis of both the semiclassi-cal master equations and the quantum shuttle dynamicsfor this configuration was performed in [46]. The authorspredict that as a result the endohedral spin-state of indi-vidual paramagnetic “shuttle” molecules such as N@C60

and P@C60 can be detected as could also very small staticforces acting on a C60-based nanomechanical shuttle.

In the fully developed self-oscillating regime the oscil-lating grain, sequentially moving electrons from one leadto the other, provides a “shuttle mechanism” for chargeas shown in Fig. 1b. In each cycle 2n electrons are trans-ferred, so the average contribution to the current fromthis shuttle mechanism is

I = 2enf , n =[CV

e+

12

], (2)

where f ≡ ω0/2π is the self-oscillation frequency. This

7

current does not depend on the tunnelling rate ωR. Thereason is that when the charge jumps to or from a lead,the grain is so close that the tunnelling rate is large com-pared to the elastic vibration frequency. Hence the shut-tle frequency — not the tunnelling rate — provides the‘bottle neck’ for this process. We emphasize that the cur-rent due to this shuttle mechanism can be substantiallylarger than the conventional current via a fixed grain.This is the case when ω0 À ωR.

To support the qualitative arguments given above wehave performed analytical and numerical analyzes basedon the simultaneous solution of Newton’s equation forthe motion of the grain’s center-of-mass and a Masterequation for the charge redistribution.

Two different approaches were developed. The first,presented in [13], gives a quantitative description of theshuttle instability for low tunnel-barrier resistances, i.e.when the rate of charge redistribution is so large (in com-parison with the vibration frequency), that the stochasticfluctuations in the grain charge during a single-vibrationperiod are unimportant. The second approach, describ-ing the opposite limit of low-charge redistribution fre-quencies characteristic of high-resistance tunnel barriers,was presented in [42].

In both cases it was shown that the electromechani-cal instability discussed above has dramatic consequencesfor the current-voltage characteristics of a single electrontransistor configuration as shown in Fig. 2. Even fora symmetric double junction, where no Coulomb stair-case appears in conventional designs, we predict that theshuttle mechanism for charge transport manifests itselfas a current jump at V = Vc and as a Coulomb staircaseas the voltage is further increased. A more precise cal-culation along the line l sketched in Fig. 2 is shown inFig. 3. The non-monotonic behavior of the current alongthis line is due to competition between the two chargetransfer mechanisms present in the system, the ordinarytunnel current and the mechanically mediated currentImech(x0, t) = δ(x(t) − x0)x(t)q(t) through some crosssection at x0. We define the shuttle current as the timeaveraged mechanical current through the plane locatedat x0 = 0. This current together with the tunnel cur-rent for the same cross section is shown in Fig. 3. As thedamping in the system is reduced the oscillation ampli-tude grows and the shuttle current is enhanced while theordinary tunnelling current is suppressed. In the limitof low damping this leads to a quantization of the totalcurrent in terms of 2ef .

The analysis presented above implicitly assumes thatthe possibly finite amplitude A of the mechanical oscil-lation is much smaller than the characteristic size of thesystem. In particular, that A does not exceed the dis-tance d between the electrodes. At the same time onemust expect anharmonic effects for large shuttle vibra-tion amplitudes. As a result the shuttle vibration fre-quency is a function of the vibration energy (and there-fore a function of the applied bias voltage). Hence theI(V ) curve given by Eq. (2) is in general more compli-

0 0.5 1 1.5 2 2.5 3 3.5

01

23

45

60

0.51

1.52

2.53

3.54

VC/e2Ω2/γν

Iπ/e

ω

l

Vc

FIG. 2: Current due to the shuttle mechanism through thecomposite Coulomb blockade system of Fig. 1. The currentis normalized to the eigenfrequency ω (in the text denotedby ω0) of elastic grain vibrations and plotted as a function ofnormalized bias voltage V and inverse damping rate γ−1 Withinfinite damping no grain oscillations occur and no Coulombstaircase can be seen. The critical voltage Vc required for thegrain to start vibrating is indicated by a line. From Ref. [13],L. Y. Gorelik et al., Phys. Rev. Lett. 80, 4526 (1998), withpermission from the American Physical Society c© 1998.

cated than the simple step-like curve shown in Fig. 2 [47].A numerical calculation reported in [48] and carried outfor parameters that are realistic for silicon-based shuttlestructures show that a shuttle instability occurs. How-ever, the resulting I(V ) characteristics contain only asingle Coulomb step.

2.1.3. Shuttling in dissipative nanostructures

From the above analysis it is clear that a large dampingis detrimental for the development of the shuttle insta-bility and in the limit where γ & ω0, elastic shuttlingof the charge becomes impossible. The mechanical la-bility of the system, however, is still a dominating fea-ture of the charge transport even in the limit of strongdissipation. The consequences of such a lability are ad-dressed in Ref. [49]. There the elastic restoring forceis assumed absent or much weaker than viscous damp-ing forces. According to that model, charge transportthrough the NEM-SET is affected both by the Coulombblockade phenomenon and the mechanical motion of thecluster. These two phenomena are coupled since thethreshold voltage for electron tunnelling depends on thejunction capacitances which, in turn, depend on the clus-ter position with respect to the leads. In general, thethreshold voltage increases when the distance betweenthe cluster and an electrode decreases.

To be specific, if a neutral cluster is located in itsequilibrium position between the electrodes no tunnellingtakes place for a bias voltage V lower than some thresh-

8

1 2 3 4 5 60

1

2

3

4

total current

shuttle current

tunnel current

Iπ/e

ω

22 /γνΩ

FIG. 3: Cross section along the line l in Fig. 2. The totaltime averaged current consist of two parts, the shuttle cur-rent and the tunneling current. The time averaged shuttlecurrent is the mechanically transferred current through thecenter of the system < δ(x(t))x(t)q(t) >, the remaining partcomes from ordinary tunneling. As the inverse damping γ−1

increase the shuttle current approaches the quantized valueIπ/eω = 3. The tunnel current is proportional to the fractionof the oscillation period spent in the middle region, |x| < λ.This fraction is inversely proportional to the oscillation ampli-tude and hence the tunnel current decreases as γ−1 increases.The fine structure in the results is due to numerical noise.From Ref. [13], L. Y. Gorelik et al., Phys. Rev. Lett. 80, 4526(1998), with permission from the American Physical Societyc© 1998.

old value V0. At V > V0 the cluster can be charged dueto tunnelling onto the cluster. At the same time, the elec-trical forces produce a mechanical displacement directedfrom the lead which has supplied the extra charge. Aftersome time the extra charge will leak to the nearest elec-trode, and the cluster becomes neutral again. An impor-tant question at this stage is if an extra tunnelling eventto the nearest electrode can take place. The answer is notevident since the electrostatic tunnelling threshold in thelast position is different from that at the initial point insystem’s center. Consequently, tunnelling to the nearestelectrode, in principle, could be suppressed due to theCoulomb blockade. The analysis made in Ref. [49] hasshown that at zero temperature there is an upper thresh-old voltage Vt below which the extra tunnelling event isnot possible. In this case the cluster is almost trappednear the electrode and the conductance is not assisted bysignificant cluster displacements between the electrodes.

For voltages above the threshold, V > Vt, there is apossibility for another tunnelling event between the grainand the nearest lead to happen after the extra charge hastunnelled off the cluster. This event changes the sign ofthe net charge on the grain. In this case the cluster canbe pushed by the Coulomb force towards the more dis-tant electrode where the above described process repeatsitself. The conductance is now assisted by significant dis-placements of the grain and this scenario is qualitatively

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1 1.5 20

0.05

0.1

I(n

A)

I(n

A)

V/V0

V/V0

FIG. 4: The solid line shows the current - voltage charac-teristics obtained by a Monte Carlo simulation of the chargetransport through a dissipation dominated system. The cal-culated DC current is plotted as a function of the bias voltageV scaled by the Coulomb blockade theshold voltage V0, thatapplies if the movable grain is equally far from both elec-trodes. The dashed line displays the current through a staticsymmetric double junction for the same parameters. For volt-ages above the threshold voltage Vt the current through thesystem increase drastically due to grain motion. The insetwhich shows a magnification of the voltage interval aroundVt. From Ref. [49], T. Nord et al., Phys. Rev. B 65, 165312(2002), with permission from the American Physical Societyc© 2002.

similar to the shuttle vibrations in fully elastic electrome-chanical structures [13]. This process is also accompaniedby a marked rise in the current through the system asshown in Fig. 4.

2.1.4. Accuracy of a mechanical single-electron shuttle.

Several mechanisms contribute to a deviation of theaverage current from the ideal shuttle value nef0:

1. Sequential electron tunnelling and co-tunnellingthrough the grain, which leads to a “shunting” DCcurrent unrelated to any grain motion.

2. Insufficient contact time t0 at the trajectory turn-ing point compared to the charge relaxation timeω−1

R . The relation between these characteristictimes determines if the grain can be fully loadedduring a single contact event.

3. Thermal fluctuations, which lift the Coulombblockade limitation for the transferred charge to bean integer number of electron charges.

The contribution of the shunting tunnelling seems to bemuch smaller than the current conveyed by a shuttling

9

grain. Indeed, the former is limited by the maximumtunnel resistance which is exponentially large. The sec-ond and third limitations to the accuracy of the shuttlingcurrent were considered by Weiss and Zwerger [50] wherea Master equation for the charge of the moving grain wasanalyzed. In this approach the shuttling was mapped ona sequence of contact events when charge transfer takesplace. The results of an analytical treatment of a simplemodel and of a numerical treatment are shown in Figs. 5and 6 taken from Ref. [50]. In Fig. 5 the average numberof electrons transferred per period, as well as the rootmean square fluctuations, are shown for T = 0. In thisfigure t0 represent the effective time the grain spendsin contact with the leads while τ denotes the RC-timeat the point of closest approach to the leads. Assum-ing that the grain is closest to the leads at a time tmax,one has τ = R(tmax)C(tmax). Thermal smearing of the

0

1

2

3

4

0 0.5 1 1.5 2 2.5

<N

>

VCL(tmax)/e

t0/τ=1t0/τ=2

t0/τ=10t0/τ=100

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5

∆N

VCL(tmax)/e

t0/τ=1t0/τ=2

t0/τ=10

FIG. 5: Average number of electrons transferred per period(upper panel) and the root mean fluctuations (lower panel)for T = 0. A Coulomb bl ockade is clearly visible: up to acritical voltage, V CL(tmax)/e = 1/2), no electrons are trans-ferred. From Ref. [50], C. Weiss and W. Zwerger, Europhys.Lett. 47, 97 (1999), with permission from the EDP Sciencesc© 1999.

single-electron shuttling is demonstrated in Fig. 6. It isclear that for t0 À τ and T = 0 the Coulomb staircase isperfect. For increasing temperatures, the Coulomb stair-case is washed out leading to an Ohmic behavior at high

0

1

2

3

4

0 0.5 1 1.5 2 2.5

<N

>

VCL(tmax)/e

T=0kBT=0.1EckBT=0.2EckBT=0.4Ec

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5

∆N

VCL(tmax)/e

kBT=0.1EckBT=0.2EckBT=0.4Ec

FIG. 6: Thermal smearing of single-electron shuttling. Theenergy scale is given by EC = e2/2C. tmax is the instantwhen a particle is located at a turning point. From Ref. [50],C. Weiss and W. Zwerger, Europhys. Lett. 47, 97 (1999),with permission from the EDP Sciences c© 1999.

temperatures.

2.1.5. Gate voltage control of shuttle mechanics

The electromechanical coupling also dramaticallychanges the transistor effect in a NEM-SET as comparedto an ordinary SET. We will discuss this problem fol-lowing Nishiguchi [51]. Let us assume that the tun-nelling can take place only between the grain and theleads, while there is no tunnelling exchange with the gate.The gate voltage controls the equilibrium position of thegrain with respect to the leads since it determines theextra charge of the grain. Schematic configuration of arelevant NEM-SET is sketched in Fig. 7. The picturedescribes a situation when the grain has a net negativecharge. The net negative charging of the grain occurs fora certain relation between the bias, Vb, and the gate, Vg,voltages. The positively charged gate electrostatically in-duces a negative charge on the grain, which tends to becompensated by the tunnelling of positive charge fromthe right lead. Here we assume that the tunnelling fromthe negatively charged remote electrode is exponentially

10

FIG. 7: Schematic configuration of a nanoelectromechanicalsingle-electron transistor (NEM-SET). From Ref. [40], R. I.Shekhter et al., J. Phys: Condens. Matter 15, R441 (2003),with permission from the Institute of Physics and IOP Pub-lishing Limited c© 2003).

suppressed. Since the grain is shifted from the centralposition, the current through the device is exponentiallysmall. An increase in the bias voltage decreases the to-tal negative charge. When the compensation is completethe grain returns to the central position restoring thetunnelling transport through the device. The “phase di-agram”in the Vg−Vb plane obtained in Ref. [51] is shownin Fig. 8. It is worth mentioning that there is a quali-tative difference between this “butterfly” phase-diagramand the “diamond” phase diagram in conventional SETdevices [39], where the mechanically blocked SET oper-ation is absent.

2.1.6. Nanoparticle chains

We continue this Section on classical shuttling of elec-trons by considering theoretical work by Nishiguchi [52]regarding nanoparticle chains. Nanoparticle chains con-sist of small metal grains stabilized by ligands, with elec-tronic transport occurring via tunnelling between themetal particles. Because of the relative softness of theligand matrix, vibrations of the metal grains can signifi-cantly modify the electronic tunnelling rates. In systemswith several degrees of freedom, the electro-mechanicalinstability at special values of system parameters canlead to excitation of more than one mode. As result,the mechanical motion becomes in general non-periodicwith a possibility of a telegraph-like switching betweenthe modes. A crossover from a periodic to a quasiperi-odic motion, as well as telegraph-like switching betweenthese regimes was demonstrated in Refs. [52, 53], whereelectron shuttling through a system of two elastically andelectrically coupled particles was numerically simulated.A telegraph-like switching was observed at some value ofbias voltage.

-3

-2

-1

0

1

2

3

G

-6 -4 -2 0 2 4 6

1

1

2

2

2

3

3

3

3

4

4

4

4

4

5

5

5

5

5

-1

-1

-2

-2

-2

-3

-3

-3

-3

-4

-4

-4

-4

-4

-5

-5

-5

-5

-5

5-5

V

V

FIG. 8: Diagram of the current I in the bias voltage-gate volt-age plane (V and VG, respectively) for symmetrically appliedsource-drain voltage. The shaded diamonds denote the volt-age regions where electron tunneling is prohibited at T = 0.The white areas correspond to mechanically suppressed SEToperation. From Ref. [51], N. Nishiguchi, Phys. Rev. B 65,035403 (2001), with permission from the American PhysicalSociety c© 2001.

2.1.7. Charge shuttling by a suspended nanotube

The case considered above of a chain of mechanicallyvibrating dots raises the more general question of thepossibility of a current induced instability in extendedmechanical objects such as long molecules and nanowires.The main difference compared to other shuttle systemsis the existence of more than one mechanical degree offreedom (mode of vibration), each of them coupled toelectron tunneling events. The nanoelectromechanicalresponse of the system to such a coupling may in gen-eral be quite complicated. The question of whether itis possible to selectively “amplify” a particular mechani-cal mode is of special fundamental and practical interest.In Ref. [54] a particular example of a multi-mode sys-tem was considered. A suspended nanowire experiencingbending mode vibrations was probed by injecting a tun-nel current through an STM tip (see. Fig. 9). This studywas motivated by an experiment [12] where inelastic elec-tron tunneling accompanied by excitation of mechanicalradial vibrations was detected. In [54] a shuttle-like in-stability resulting in the onset of bending vibrations of afinite amplitude was theoretically predicted to occur asa result of the injection of current. Due to the differentcoupling strengths between the electronic subsystem and

11

FIG. 9: Injection of electrical charge in a nanotube by a STMtip.

the various bending modes it was shown that an insta-bility of a specific mode could be selectively induced byvarying the applied bias voltage [54]. According to theestimation in Ref. [55] the predicted nanomechanical in-stability should be observable under conditions similarto those at hand in the experiment [12]. However, it isimportant that the tunnel resistances between the STMtip and the nanotube on the one hand and between thenanotube and the electrodes on the other, do not differsignificantly.

2.2. The charge shuttle as a nanomechanicalratchet

One advantage of the self-oscillating shuttle structuresis that it is possible to use a static bias voltage to gener-ate mechanical oscillations of a very high frequency. Thefact that the system has an intrinsic and stable oscillat-ing mode suggests that the application of an oscillatingvoltage may lead to new interesting effects related to theinterplay between the external AC drive and the inter-nal frequency of the device. Moreover, as the dynamicsof shuttle systems is essentially nonlinear, this interplayshould emerge in a wide interval of the ratio of the twofrequencies.

A shuttle driven by a time-dependent applied bias wasconsidered by Pistolesi and Fazio [56]. They showed thatan asymmetric structure can act as a ratchet (see, e.g.,Ref. [57]) in which the forcing potential is generated ina self-consistent way. A sizeable ratchet effect is presentdown to bias frequencies much smaller than the mechan-ical resonance frequency ω0, due to the adiabatic changeof the equilibrium position of the grain. In a recent ex-periment [58] some results similar to the predictions ofRef. [56] were observed. However, in this experiment thesingle-electron tunnelling limit was not reached.

Assuming a harmonic bias voltage, V = V0 sinωt, theauthors of Ref. [56] simulate the grain dynamics andstochastic electron transfer for an asymmetric systemcharacterized by the resistances RL(0) 6= RR(0). Theasymmetry is essential, since no DC current can be gen-erated in a symmetric device. As the electron transi-tion rates are proportional to |V (t)|, they turn out to be

time-dependent. According to the simulation, the sys-tem reaches a stationary behaviour after a transient time.Figure 10 shows the stationary DC current as a functionof the external bias frequency ω.

FIG. 10: Current through a shuttle system as a function ofbias voltage frequency for given bias voltage amplitude anddevice parameters (ε = 0.5, γ/ω0 = 0.05 and Γ/ω0 = 1,see text and Ref. [56]. The result of the simulation of thestochastic dynamics (points) is compared with an approxi-mate analytical solution for the current (full line). In thesmall frequency region, enlarged in the upper inset, severalresonances at fractional values of ω/ω0 appear. The dottedline is an analytical result in the adiabatic limit. Lower inset:current noise from the simulation (points) and analytical re-sult (dashed) for the static SET. From Ref. [56], F. Pistolesiand R. Fazio, cond-mat/0408056 (unpublished).

The rich structure shown in the figure is generic; quali-tatively identical behaviour was observed for a wide rangeof parameters. The existence of a direct current as aresult of an applied periodic modulation shows that thecharge shuttle behaves as a ratchet [57]. Since the systemis nonlinear, the external driving force affects the dynam-ics also for values of ω very different from the intrinsicfrequency ω0. Note that in this model the nonlinearitiesare intrinsic to the shuttle mechanism. They are due tothe specific time dependence of the grain charge, en(t),rather than to a nonlinear mechanical force. As is evi-dent from Fig. 10 the ratchet behaviour is present alsoin the adiabatic limit, ω/ω0 ¿ 1. In addition, a series ofresonances appear. These are due to frequency lockingwhen ω/ω0 = q/p and q and p are integers, see, e. g., [59].In this case the motion of the shuttle and the oscillatingsource become synchronized in such a way that duringq periods of the oscillating field the shuttle performs poscillations.

In contrast to the ratchet behaviour [57], where anac voltage induced a nonzero dc shuttle current only inan asymmetric tunnel junction, an ac generated dc cur-rent was predicted to occur in a symmetric double-shuttlestructure in [60]. Such a system, where two shuttles arecoupled in series double, exhibits a parametric instability

12

that induces both shuttles to vibrate in such a way thatthe distance between them oscillates. In contrast to thesingle-shuttle structure studied in [57], where the shut-tle resistance oscillates with twice the frequency of theshuttle motion itself, in double-shuttle structure mutualresistance oscillates with the same frequency as the shut-tles. This difference alters qualitatively the rectificationability of the system. As a result a rectification effect isapparently possible in a fully symmetric device [60]. Therectified current can flow in either direction dependingon how the parametric instability has developed.

2.3. Classical shuttling of electron waves

In this Section we follow the considerations of Fedoretset al. [43]. Here it is assumed that a vibrating grain hasa single resonant level, both the position x(t) of this leveland the coupling of the grain to the leads, TL,R(t), areoscillatory functions of time, see Fig. 11. The effective

x

µL = eV

µR = 0

LeadLead

Dot

FIG. 11: Model system consisting of a movable quantum dotplaced between two leads. An effective elastic force acting onthe dot from the leads is described by the parabolic potential.Only one single electron state is available in the dot and thenon-interacting electrons in each have a constant density ofstates. From Ref. [43], D. Fedorets et al., Europhys. Lett. 5899 (2002), with permission from the EDP Sciences c© 2002.

Hamiltonian of the problem is defined as

H =∑

α,k

(εαk − µα)a†αkaαk (3)

+εd(t)c†c +∑

α,k

Tα(t)(a†αkc + c†aαk) .

Here TL,R = τL,R exp∓x(t)/2λ is the position-dependent tunneling matrix element, εd(t) = ε0 − eEx(t)is the energy level in the dot shifted due to the voltageinduced electric field E , a†αk creates an electron with mo-mentum k in the corresponding lead, α = L,R is the leadindex, c† creates an electron in the dot. The first term inthe Hamiltonian describes the electrons in the leads, thesecond corresponds to the movable quantum dot and thelast term describes tunnelling between the leads and thedot.

The evolution of the electronic subsystem is deter-mined by the Liouville-von Neumann equation for thestatistical operator ρ(t),

i∂tρ(t) = ~−1[H, ρ(t)]− , (4)

while the center-of-mass motion of the dot is governed byNewton’s equation,

x + w20x = F/m . (5)

Here ω0 =√

k0/m, m is the mass of the grain, k0 is aconstant characterizing the strength of the harmonic po-tential, and F (t) = −Tr

[ρ(t) ∂H/∂x

]. The force F is

computed from an exact solution of the tunnelling prob-lem, which exists for arbitrary Tα(t) and εd(t). Using theKeldysh Green’s function approach [61] in the so-calledwide band approximation and following Ref. [62] one ob-tains the force F as

F (t) =∑α

gα

∫dε fα(ε)

eE |Bα(ε, t)|2

+(−1)αλ−1Tα(t)Re[Bα(ε, t)]

, (6)

where

Bα(ε, t) = −i~−1

∫ t

−∞dt1Tα(t1)

× exp

i~−1

∫ t

t1

dt2

[ε− εd(t2) + i

Γ(t2)2

],

and Γ(t) = 2π∑

α gα|Tα(t)|2, gα is the densityof states in the corresponding lead while fα(ε) =exp[β(ε− µα)] + 1−1. The first item in the expres-sion (6) for F (t) is the electric force due to accumulatedcharge; the second item is the “cohesive” force due toposition-dependent hybridization of the electronic statesof the grain and the leads. Equations (5) and (6) canbe used to study the stability of the equilibrium posi-tion of the cluster. By linearizing Eq. (5) with respectto the small displacement ∝ e−iωt and solving the properequations of motion, one can obtain a complex vibrationfrequency. The mechanical instability can then be de-termined from the inequality Im ω ≥ 0. As shown inRef. [43], the instability occurs if the driving bias voltageexceeds some critical value, which for a symmetrically ap-plied voltage is eVc = 2(ε0 +~ω0). It was also shown thatin the limit of weak electromechanical coupling, whenΓ/(4mω2

0λ2) and 2eEλ/(√~2ω2

0 + Γ2) ¿ 1, the instabil-ity develops into a limit cycle. This is in contrast with aclassical shuttle, where stability of the system could beachieved only at finite mechanical dissipation. The rea-son for the difference is that here dissipation is providedby an explicit coupling to the electronic degrees of free-dom in the Hamiltonian (3). In the classical treatmentdissipation was only described phenomenologically by adamping constant in Newton’s equation. One can seea qualitative agreement between the experimentally ob-served I(V ) curve for a fullerene based NEM-SET shown

13

in Fig. 18 and the results of the above calculation dis-played in Fig. 12. However, there are alternative inter-

0 10 20 30 40 50 600

0.05

0.1

0.15

eV, meV

I, nA

χ = 0.20 nm−1

χ = 0.15 nm−1

χ = 0.10 nm−1

FIG. 12: I-V curves for different values of the parameter χwhich characterizes the strength of the electric field betweenthe leads for a given voltage: ~ω0 = 5 meV, T = 0.13 meV,ε0 = 6 meV, and Γ = 2.3 µeV. Best fit to Ref. [6] is obtainedfor an asymmetric coupling to the leads; here we use ΓR/ΓL =9. From Ref. [43], D. Fedorets et al., Europhys. Lett. 58 99(2002), with permission from the EDP Sciences c© 2002.

pretations of the experiment by Park et al. [6] (see [63–67]), which are based on quantum mechanical treatmentsof the mechanical motion. Some of these will be discussedbelow.

The authors of Refs. [68–70] have performed numericalsimulations of current induced nanovibrations in fullerenebased tunneling device (Au-C60-Au double junctions)based on an ab initio nonequilibrium formalism. Theirmethod involves the use of Keldysh Green’s functionsfor describing the nonequilibrium electron subsystem andquantum mechanics for solving the fullerene dynamics.The coupling between the mechanical and electronic sub-systems was treated in the framework of time-dependentscattering theory. Using parameters relevant to the ex-periment [6] they demonstrate that by applying a dc biasvoltage a time dependent current is generated. The au-thors interpreted this as the result of a nanoelectrome-chanical instability similar to the shuttle instability dis-cussed above. Special features of the dynamics in thiscase appear because of the resonant character of the elec-tron tunneling through the C60 molecule (see also [71]).Similar simulations for a nanojunction, containing a C20

molecule were performed in [72]. Here, in contrast towhat is done for conventional shuttle systems, the limitof strong tunneling coupling at the Au-C20 interface wasconsidered and no shuttle instability was detected. In-stead a pronounced step in the differential conductanceversus voltage was detected indicative of the strong ef-fect of vibron-assisted tunneling of electrons through themolecule.

2.4. Charge transfer through a quantum oscillator

In the previous discussion we assumed that the me-chanical degree of freedom was classical. As discussed inthe beginning of Sec. 2 this assumption is not always cor-rect. For an oscillator with a small enough mass, e.g., thequantum mechanical zero-point vibration amplitude, x0,can be comparable with the the center-of-mass displace-ment. In this case electronic and mechanical degrees offreedom behave as coupled parts of a quantum systemwhich in its entirety should be described by quantummechanics. The coupling between the subsystems is pro-vided by (i) the dependence of electronic levels, charges,their images, and, consequently of the electric forces onthe mechanical degrees of freedom, and by (ii) the depen-dence of tunnelling barriers on the spatial configurationof the system.

In general, charge transfer in the systems under consid-eration is assisted by excitation of vibrational degrees offreedom which is similar to phonon-assisted tunnellingthrough nanostructures. However, in some cases thecenter-of-mass mechanical mode turns out to be morestrongly coupled to the charge transfer than other modes.In this case one can interpret the center-of-mass-mode as-sisted charge transfer as quantum shuttle transport, pro-vided that the center-of-mass motion is correlated in timewith the charge on the dot (see the definition of shuttletransport in the introduction). Several authors have ad-dressed the charge transport through a quantum oscil-lator, and several models have been discussed. Some ofthis work is reviewed below.

Single-electron tunnelling through molecular struc-tures under the influence of nanomechanical excitationshas been considered by Boese and Schoeller [63]. The au-thors developed a quantum mechanical model of electrontunnelling through a vibrating molecule and used it tomodel the experiment described in Ref. [6]. In contrastto what was done in Secs. 2.1, 2.3 and Refs. [13, 43], theyassumed the vibrational frequency to be several ordersof magnitude larger than the electron tunnelling rates.Their system is described by an effective Hamiltonian in-volving local bosonic degrees of freedom of the molecule.One of the local modes can be interpreted as describingcenter-of-mass motion. The quantum mechanical calcu-lation leads to a set of horizontal plateaus in the I − Vcurve due to excitation of different vibrational modes. Itis shown that in some regions of parameter space a nega-tive differential resistance occurs. A similar calculation,but with a more detailed account of the dependence ofthe charge-transfer matrix elements on the shuttle coor-dinate, has been published by McCarthy et al. [64].

References [63] and [64] provide a qualitative explana-tion of the experiment by Park et al. [6]. Other fullyquantum-mechanical efforts to model this experiment in-volving vibron-assisted tunnelling rather than the shuttlemechanism can be found in Refs. [65–67]. These are ac-tually alternative explanations to that given in Ref. [43],where the center-of-mass motion was treated classically.

14

Vibron-assisted tunnelling is the appropriate picture forhigh damping rates (low Q-factor) and classical shuttlingfor low damping (high Q-factor). To determine the Q-factor relevant for the experiment of Park et al. [6] re-quires additional information, e.g., from noise measure-ments, see Sec. 5. It is worth mentioning that the shuttlemodel predicts a finite slope of the I − V curve at largevoltages, which is more similar to the experimental re-sult.

A quantum oscillator consisting of a dot coupled bysprings to two flanking stationary dots attached to semi-infinite leads was considered by MacKinnon and Ar-mour [73]. These authors concentrate on the quantumaspects of the dot and electron motion. It is shown thatthe I−V characteristics of the model shuttle can largelybe understood by analyzing the eigenspectrum of the iso-lated system of three dots and the quantum oscillator.Tunnelling coupling of the dot states, to each other andto the position of the oscillator, leads to repulsion of theeigenvalues and mixing of the eigenstates associated withstates localized on individual dots. The mixed states con-sist of superpositions of the states associated with theindividual dots and hence lead to delocalization of theelectronic states between the dots. Analysis of the cur-rent which flows when the shuttle is weakly coupled toleads, reveals strong resonances corresponding to the oc-currence of the delocalized states. The current throughthe shuttle is found to depend sensitively on the amountby which the oscillator is damped, the strength of thecouplings between the dots and the background temper-ature. When the electron tunnelling length λ is of orderx0, current far from the electronic resonance is dominatedby electrons hopping on and off the central dot sequen-tially. As the authors state, then the oscillator can beregarded as shuttling electrons across the system as hasbeen discussed in Sec. 2.1.

In the already mentioned work described in Refs. [63–67, 73], as well as in some other papers [65–67, 74–76]that deal with various aspects of the NEM-SET devicein the regime of quantized mechanical grain motion, noshuttle instability was found. The reason is that eitherthe coupling between electron tunnelling and mechanicalvibrations is ignored [63], or that strong dephasing of themechanical dynamics was assumed [65–67, 74–76]. How-ever, the analysis of the shuttle instability of a NEM-SETstructure in the quantum regime is important. Indeed,at the initial stage of the instability the oscillation ampli-tude, A, can be of the same order as the the zero-pointoscillation amplitude, x0. Therefore quantum fluctua-tions can be important. The necessary analysis of thisissue has been performed by Fedorets et al. [77] and, inmore detail, by Novotny et al. [78] and Fedorets et al. [79].In these papers it was assumed that the shuttling grainhas only one electron level, which can be either empty oroccupied. The interaction Hamiltonian included linearcoupling to the electrical field E , (eE xn, where n is theelectron number operator for electrons on the grain), aswell as a tunnelling coupling to the leads that is expo-

nentially dependent on the displacement operator x. Inaddition, an interaction of the oscillator with an externalthermal bath was included and treated to lowest order ofperturbation theory. By projecting out the leads and thethermal bath a so-called generalized master equation forthe density matrix of the system involving electron statesand oscillator variables is obtained under the conditionthat eV À ~ω0, |ε0 − µ|, kBT . In Ref. [78], this equationis numerically analyzed for the case x0 ∼ λ. The authorssuggest to visualize the quantum features expressing theresults through the Wigner distribution functions,

Wii(x, p) =∫ ∞

−∞

dy

2π~

⟨x− y

2

∣∣∣∣ρ∞ii∣∣∣∣x +

y

2

⟩eipy/~ . (7)

Here ρ∞ii is the stationary density matrix, ρii(t → ∞),which is diagonal in the electron states i = 0, 1. The clas-sical motion of the oscillator can be characterized by itsphase trajectory which is a sharp line in the x, p-plane,given by the relation p2/2m+kx2/2 = const. Smearing ofthis line by quantum fluctuations shows the importanceof quantum effects. The results of the numerical analy-sis of Wii(x, p) for λ = x0 and different frictions (γ) areshown in Fig. 13. For large friction (γ = 0.25) the Wigner

FIG. 13: Phase space picture of the tunnelling-to-shuttlingcrossover. The respective rows show the Wigner distributionfunctions for the uncharged (W00) and charged (W11) statesof the oscillator as well as the sum of these states (Wtot) inphase space (horizontal axis - spatial coordinate in units ofx0, vertical axis - momentum in units of ~/x0). Parametervalues used are λ = x0, d ≡ eE/mω0 = 0.5x0, Γ = 0.05~ω0.The values of γ are in units of ~ω0. The Wigner functions arenormalized within each column. From Ref. [78], T. Novotnyet al., Phys. Rev. Lett. 90, 256801 (2003), with permissionfrom the American Physical Society c© 2003.

functions are smeared around the origin (or around ashifted origin when charged) to an extent that dependson the zero-point vibration amplitudes with no particu-lar correlations between its charge state and momentum.This means that the state of the grain is very close tothe oscillator ground state, i.e. charge transfer excites

15

the mechanical degree of freedom to a very small extent.This is consistent with the vibrationally assisted charge-transfer mode. For small friction (γ = 0.05), on the otherhand, the Wigner functions are smeared around the clas-sical phase trajectory p2/2m+kx2 = hω0 (ringlike shapeof Wtot with a hole around the origin), and the correla-tion between the charge state and the mechanical motionis very strong (half-moon shapes of W00, W11) Thus onecan see a quantum precursor of the electronic shuttle, seealso Ref. [77]. The classical shuttle picture is expectedto emerge in the quasiclassical limit d, λ À x0, whered ≡ eE/mω2

0 is a measure of the electric field strength.Interestingly, there exist a region where both vibronic

assisted charge transfer and shuttle features are present.In the crossover region (intermediate friction, γ = 0.1) wecan see that both regimes of transport are contributingadditively (ringlike shape plus an incoherent peak aroundthe origin of Wtot).

In Ref. [79], an analytical treatment of the case x0 ∼d ¿ λ is reported, and an intermediate regime betweenvibronic assisted charge transfer and shuttling was found.The condition x0 ¿ λ allows one to linearize the problemin the displacement x for low levels of excitation of themechanical degree of freedom. By doing so one findsa closed set of equations describing the time evolutionof 〈x〉 and 〈p〉. It is found that 〈x〉 and 〈p〉 increaseexponentially with time, i.e., the vibrational ground stateis unstable above a threshold value of the electric field,

E > Eth =γ

Γω2

0λ

e.

In this way the development of the shuttle instabilityfrom quantum fluctuations is understood.

The exponential increase of the vibration amplitudeleads the system into a nonlinear regime, where the sys-tem reaches a stable shuttling state. This state was stud-ied using the Wigner density as suggested in Ref. [78].

The global behavior of the Wigner density in the shut-tling state depends on the electric field, E . What isparticularly interesting is that there are two shuttlingregimes — a classical regime and a quantum regime. Theboundary between these regimes is given by the crossoverfield Eq = ce−1mω2

0λ(x0/λ)4 where c is a numerical con-stant. For typical parameters c ∼ 10−2. For weak me-chanical dissipation, γ/m . Γc(x0/λ)4, the crossoverfield Eq is larger than the threshold field Eth, so thereis the region of electrical field (Eq > E > Eth) where theshuttle regime has a specific quantum character. Thisregime is described by a Wigner function similar to theone obtained numerically in Ref. [78] for x0 = λ (seeFig. 13) but scaled by the large parameter λ/x0. It ischaracterized by pronounced quantum fluctuations andcan be interpreted as a quantum regime.

For strong electrical fields, E À Eq, the steady-stateWigner functions are only slightly smeared over the clas-sical trajectory p2/2m + k0x

2/2 = k0A2cl(E) and demon-

strate pronounced shuttle features (here Acl ∼ λ is theamplitude of the classical shuttle oscillations and k0 is

the effective spring constant). This regime can thereforebe interpreted as a classical regime.

The results of Ref. [79], reviewed above, demonstratethat the condition λ À x0, although a necessary con-dition for the shuttling regime to be classical, is not asufficient condition. This is because the appearance ofshuttle vibrations is a nonequilibrium phenomenon thatcomes from a nanomechanical instability induced by anapplied bias voltage. As a nonequilibrium phenomenonthe nature of the vibrations do not only depend on the pa-rameters of the device (such as x0 and λ), but also on theapplied voltage V and the resulting electric field E thatacts on the shuttle. There are two “channels” throughwhich the shuttle vibrations can be excited (correspond-ing to two types of forces on the shuttle). The first chan-nel is electrical in nature and is caused by the electriccharge carried by tunnelling electrons. This channel hasa classical analog - it corresponds to the work done bythe electric force on the charge accumulated on the shut-tle. The second channel is not connected with the elec-tric charge but originates from energy corrections causedby quantum delocalization of shuttle electrons onto theleads. The nature of this channel, which is active whenthe shuttle is in tunnelling contact with either lead, canbe said to be cohesive. It has no classical analog at largevoltages V applied to the system. The reason is that theclassical cohesive force acting on the shuttling grain be-comes zero in the limit when all electronic states on theleads are either completely empty or completely filled.Hence it is only quantum fluctuations that contribute avibron-assisted pumping of cohesive energy into the shut-tle. For small enough electric fields, E ¿ Eq, quantumcohesive pumping dominates and leads to the formationof non-classical, quantum shuttle vibrations with largeamplitude of order λ.

The temperature dependence of electron transportthrough a quantum shuttle was recently studied inRef. [80] assuming that the electric field is below the in-stability threshold. At low temperatures the calculatedI − V curve shows pronounced steps. It is shown thatin the classical limit, x0/λ . 0.3, the temperature de-pendence of the I − V curve is weak. However, for aquantum shuttle, x0/λ & 0.6, a variety of behaviors ispredicted. The behaviors depend on how deep the shut-tle is in the quantum regime and can vary from a 1/T -decrease to an exponential growth. It is stated that theresults can explain a variety of temperature dependenciesthat have been observed for electron transport throughlong molecules.

2.5. Spin-dependent transport of electrons in ashuttle structure

2.5.1. Nanomechanical manipulation of nanomagnets

The possibility to place transition-metal atoms or ionsinside organic molecules introduces a new “magnetic” de-

16

gree of freedom that allows the electronic spins to becoupled to mechanical and charge degrees of freedom[81]. By manipulating the interaction between the spinand external magnetic fields and/or the internal inter-action in magnetic materials, spin-controlled nanoelec-tromechanics may be achieved. An inverse phenomenon— nanomechanical manipulation of nanomagnets — wassuggested earlier by Gorelik et al. [82]. A magnetic field,by inducing the spin of electrons to rotate (precess) at acertain frequency, provides a clock for studying the shut-tle dynamics and a basis for a DC spectroscopy of thecorresponding nanomechanical vibrations. Since spin ef-fects are sensitive to an external magnetic field the elec-tron transport through a shuttle structure becomes spin-dependent.

A particularly interesting situation arises when elec-trons are shuttled between electrodes that are so-calledhalf-metals where all the electrons are in the same spinstate — the material is fully spin-polarized. Examples ofsuch materials can be found among the perovskite man-ganese oxides, a class of materials that show an intrinsic,so called “colossal magnetoresistance” effect at high mag-netic fields (of order 10-100 kOe) [83].

A large magnetoresistance effect at lower magneticfields has been observed in layered tunnel structureswhere two thin perovskite manganese oxide films are sep-arated by a tunnel barrier [83–87]. Here the spin polar-ization of electronic states crucially affects the tunnellingbetween the magnetic electrodes. Indeed, electrons ex-tracted from the source electrode have a certain spin po-larization while to be injected into a drain electrode theyhave to be polarized in a generally different direction.Clearly the tunnelling probability and hence the resis-tance must be strongly dependent on the relative orien-tation of the magnetization of the two electrodes. Therelative magnetization can be tuned by an external mag-netic field. A change in the resistance of trilayer devicesby factors of order 2-5 have in this way been induced bymagnetic fields of order 200 Oe [84–86]. The requiredfield strength is determined by the coercitivities of themagnetic layers. This makes it difficult to use a tun-nelling device of the described type for sensing very lowmagnetic fields.

A new functional principle — spin-dependent shuttlingof electrons — for low-magnetic field sensing purposeswas proposed by Gorelik et al. in Ref. [88]. This princi-ple may lead to a giant magnetoresistance effect in exter-nal fields as low as 1-10 Oe. The key idea is to use theexternal magnetic field to manipulate the spin of shut-tled electrons rather than the magnetization of the leads.Since an electron spends a relatively long time on a shut-tle, where it is decoupled from the environment even aweak magnetic field can rotate its spin by a significantangle. Such a rotation allows the spin of an electron thathas been loaded onto the shuttle from the spin-polarizedsource electrode to be reoriented in order to allow theelectron finally to tunnel from the shuttle to the spin-polarized drain lead. In this way the shuttle serves as a

very sensitive magnetoresistance (GMR) device.The model employed in [88] assumes that the source

and drain are fully polarized in opposite directions. Amechanically movable quantum dot (described by a time-dependent displacement x(t)), where a single energy levelis available for electrons, performs driven harmonic oscil-lations between the leads. The external magnetic field,H, is perpendicular both to the orientations of the mag-netization in both leads and direction of mechanical mo-tion.

The spin-dependent part of the Hamiltonian is speci-fied as

Hs(t) = J(t)(a†↑a↑ − a†↓a↓)−gµH

2(a†↑a↓ + a†↓a↑) ,

where J(t) = JR(t) − JL(t), a†↑(↓)(a↑(↓)) are the cre-ation (annihilation) operators on the dot, JL(R)(t) ≡JL(R)[x(t)] are the exchange interactions between the on-grain electron and the left (right) lead, g is the gyromag-netic ratio and µ is the Bohr magneton. The properLiouville-von Neumann equation for the density matrixis analyzed and an average electrical current is calculatedfor the case of large bias voltage.

The behavior of the current depends on an interplaybetween three frequency scales: (i) the frequency of spinrotation, determined by the tunnel exchange interaction,JL(R), with the magnetic leads; (ii) the frequency of spinrotation in the external magnetic field, gµH/~, and (iii)the frequency of shuttle vibrations, ω0 .

In the limit of weak exchange interaction, Jmax ¿ µHone may neglect the influence of the magnetic leads onthe on-dot electron spin dynamics. The resulting currentis

I =ω0e

π

sin2(ϑ/2) tanh(w/4)sin2(ϑ/2) + tanh2(w/4)

, (8)

where w is the total tunnelling probability during thecontact time t0, while

ϑ = ~−1gµH(π/ω − t0) ∼ πgµH/~ω0

is the rotation angle of the spin during the “free-motion”time. The H-dependence of magneto-transmittance inthis limit is characterized by two different scales. Thefirst one is associated with resonant magnetic field depen-dence through the angle ϑ in the denominator of Eq. (8).This scale is

δH = w~ω0/gµ , (9)

The second scale,

∆H = ~ω0/gµ , (10)

comes from the periodic function sin2(ϑ/2) in the numer-ator of Eq. (8). The magnetic-field dependence of thecurrent is presented in Fig. 14,a. Dips in the transmit-tance of width δH appear periodically as the magnetic

17

FIG. 14: Magnetic-field dependence of the transmittance ofthe shuttle device for the limiting cases of a) weak and b)strong exchange coupling between dot and leads. The period∆H and the width δH of the ”dips” are given by Eqs. (9) and(10) for case a) and δH is given by Eq. (11) for the case b).Adapted from [88], L. Y. Gorelik et al., cond-mat/0402284(unpublished), Phys. Rev. B (in press) with permission fromthe American Physical Society c© 200?.

field is varied, the period being ∆H. This amounts toa giant magneto-transmittance effect. It is interestingto notice that by measuring the period of the variationsin I(H) one can in principle determine the shuttle vi-bration frequency. This amounts to a DC method forspectroscopy of the nanomechanical vibrations. Equa-tion (10) gives a simple relation between the vibrationfrequency and the period of the current variations. Thephysical meaning of this relation is very simple: everytime when ω/Ω = n+1/2 (Ω = gµH/~ is the spin preces-sion frequency in a magnetic field) the shuttled electronis able to fully flip its spin to remove the ”spin-blockade”of tunnelling between spin polarized leads having theirmagnetization in opposite directions.

A strong magnetic coupling to the leads, Jmax À gµH,preserves the electron spin polarization, preventing spin-flips of shuttled electrons due to an external magneticfield. However, if the magnetization of the two leadspoint in opposite directions, the signs of the exchangecoupling to the leads are different. Therefore, the ex-change couplings to the two leads tend to cancel outwhen the dot is in the middle of the junction. Hence,the effective exchange Hamiltonian affecting a dot elec-tron periodically changes sign, being small close to thetime of sign reversal, see Fig. 15. Thus the effect of anexternal magnetic field is negligible almost everywhere,except in the vicinity of the level crossing, where the ex-ternal magnetic field removes the degeneracy forming agap is in the spectrum (dashed curve).

The electronic spin-flip in this case occurs via aLandau-Zener [89] reflection from the gap. Note thatin this case a Landau-Zener transition across the gap

FIG. 15: On-dot energy levels for spin-up and spin-down elec-tron states as a function of the position of the dot. Levelcrossing in the middle of the device is removed by an externalmagnetic field. From Ref. [88], L. Y. Gorelik et al., cond-mat/0402284 (unpublished), Phys. Rev. B (in press) with per-mission from the American Physical Society c© 200?.

is a mechanism for backscattering of the electron, sincethis is the channel where the electronic spin is preserved.The schematic I(H) dependence for this case is shownin Fig. 14. The width δH of the minimum in the I(H)dependence is

δH =√

Jmin~ωπgµ

, (11)

where Jmin = min(JL(R)(t)).Thus the shuttling of spin-polarized electrons can fa-

cilitate a giant magneto-transmittance effect caused byshuttling of spin-polarized electrons between magneticsource and drain. A typical estimate for the magneticfield leading to a pronounced effect is δH ' 1÷ 10 Oe.

2.5.2. Spintronics of a nanoelectromechanical shuttle

In Sec. 2.5.1 we showed that the charge transferthrough a nanomagnetic shuttle structure can be verysensitive to an external magnetic field. This sensitivitybrings a possibility to trigger the shuttle instability byrelatively weak magnetic fields. Below we discuss themagnetic-field-induced shuttle instability [90], which canoccur in structures with spin-polarized electrons. Thesource of the effect is an influence of the magnetic fieldon the spin-dependent electron transfer mediated by me-chanical vibration of a movable grain.

Let us consider the same system as discussed in theprevious Section. This system resembles the experimen-tal setup of Ref. [91]. A movable grain having two elec-tronic states with opposite spin directions is placed be-tween magnetic leads having equal and oppositely di-rected magnetizations able to provide full spin polariza-tions of electrons. An external magnetic field is assumedto be oriented perpendicularly to the magnetizations ofthe leads.

The shuttle instability was studied in Sec. 2.4 by com-bining the vibrational dynamics of the oscillating grainwith the quantum dynamics of the transferred electron.The generalization of the procedure outlined in Sec. 2.4

18

amounts to introducing a spin-dependent Wigner distri-bution, Wσσ′(x, p), defined as

Wσσ′(x, p) =∫ ∞

−∞

dy

2π~

⟨x− y

2

∣∣∣∣ρσσ′

∣∣∣∣x +y

2

⟩eipy/~ ,

ρσσ′ = (1/2) (ρ0δn,0 + ρ2δn,2)δσσ′ + ρσσ′1 δn,1 .

Here ρ0,2 are operators in vibrational space while ρσσ′1

works in both vibrational and spin spaces. In [90], equa-tions for the time evolution of Wσσ′(x, p) were derivedand their stationary solutions were analyzed in the limitof weak electromechanical coupling and for the quasi-classical regime, i. e. when the tunnelling length λ ismuch larger than the zero-point oscillation amplitude x0.Various stationary regimes can be identified by a “phase

FIG. 16: Regimes of a quantum oscillator depending on elec-trical and magnetic fields for Γ/ω0 = 0.1. E0 = γλω0/e,H0 = ~ω0/µ, Hc = (

√3/2)~Γ/µ. .

diagram” in the (E ,H)-plane, Fig. 16. The three do-mains in this picture correspond to three different typesof behavior of the nanomechanical oscillator.

In the “vibronic” domain (v) the system is stable withrespect to mechanical displacements from the equilibriumposition. The “shuttle” domain (s) corresponds to de-veloped mechanical vibrations behaving classically. Thethird stationary regime is a “mixed” domain (m). Thisdomain appears because the v- and s-regimes becomeunstable at different values of E and H provided H ex-ceeds some value Hc = (

√3/2)~Γ/µ. While the shuttle

regime becomes unstable below the line v, the “vibronic”state becomes unstable above the line “s”. Between theselines (m-domain) both states are stable, and the oscil-lator “bounces” between the v and s type of behavior.This bouncing is due to random electric forces caused bystochastic variations of the grain charge occurring in thecourse of tunnelling events. The transition time betweenthe two locally stable regimes, τv↔s, depends on the elec-tromechanical coupling parameter ε = d/λ ¿ 1, and can

be expressed as

τv↔s = ω−10 exp

(Sv↔s

ε

).

Here Sv↔s(E ,H) ∼ 1 is a function of the external fields.Since ε ¿ 1, at Sv→s 6= Ss→v the switching rates cor-responding to the reverse transitions differ exponentiallyleading to an exponential difference in realization prob-ability for the two regimes. The line p is determinedby the equality Sv→s = Sv←s, i. e. it corresponds toequal rates of the v → s and reverse transitions. Be-low this line, the probability of the v-regime is exponen-tially larger comparing to the s-regime, while above thisline the s-regime exponentially dominates. Due to thesmallness of the electromechanical coupling, ε ¿ 1, thetransition between the two regimes is very sharp. Hencethe change of vibration regimes can be regarded as a“phase transition”. Such a transition will manifest it-self if the external fields are changed adiabatically onthe time scale maxτs↔v. One can expect enhancedlow-frequency noise, ω . τ−1

s↔v, around the line p as ahallmark of the transition.

In the opposite limit, either s- or v- regimes can be“frozen’ in the mixed domain while crossing the line p.If one starts in the v-regime it preserves until the systemcrosses the line s, and if one starts from the s-regime itpreserves until the system crosses the line v. Hence, oneshould observe a hysteretic behavior of the non-adiabaticshuttle transition.

Two different scenarios for the onset of shuttle vibra-tions were demonstrated in Ref. [90]. If one crosses overfrom the v- to the s-region when H < Hc i. e. avoid-ing the mixed phase, the onset is soft. In this regime,after crossing the border, the vibration amplitude growsgradually from zero to some finite value. For H > Hc

the onset is hard. In this case the vibration amplitudehas a step at the transition point, which corresponds tocrossing either the p or the s line depending on whetherthe transition is adiabatic or not.

Thus, following Fedorets et al. [90], we have demon-strated that magnetic-field-controlled spin effects lead toa very rich behavior of nanomechanical systems.

3. EXPERIMENTS ON ELECTRONSHUTTLING

A recent experimental realization of the shuttle insta-bility resulting in a classical shuttle-electron transfer wasreported by Tuominen et al. [41]. The experimental setupis shown in Fig. 17. The measured current-voltage char-acteristics display distinctive jumps and hysteresis whichreflect the influence of the vibrational environment (themetal beam the figure) on the shuttle dynamics.

Systems in which electron transport between two con-tacts is mediated by a vibrational mode of a self-assembled structure have also been investigated [6, 92].The most striking example of such a system is the C60

19

FIG. 17: Experimental setup. The block, shuttle, and beamare made of brass. The dimensions of the beam are 40 mm× 22 mm × 1.6 mm. The shuttle mass is 0.157 g, its radius2.06 mm; the effective mass of the bending beam is 30 g,its fundamental vibrational frequency 210 Hz, and its qualityfactor 37. Measurements indicate the influence of anothervibrational mode at 340 Hz. The micrometer is used to adjustthe shuttle gap d. The natural pendulum frequency of thesuspended shuttle is 2.5 Hz. From Ref. [41], M. T. Tuominenet al., Phys. Rev. Lett. 83, 3025 (1999) with permission fromthe American Physical Society c© 1999.

single electron transistor, fabricated by Park et al. [6].In this device, a single C60 molecule was deposited in anarrow gap between gold electrodes. The current flowingthrough the device was found to increase sharply when-ever the applied voltage was sufficient to excite vibrationsof the molecule about the minima of the van der Waalspotential in which the molecule resides, or an internalmode of the molecule itself.

These transport measurements provided clear evidencefor coupling between the center-of-mass motion of theC60 molecules and single-electron hopping. This newconduction mechanism had not been observed previ-ously in quantum dot studies. The coupling is man-ifested as quantized nanomechanical oscillations of theC60 molecule against the gold surface, with a frequencyof about 1.2 THz. This value is in good agreement witha simple theoretical estimate based on van der Waalsand electrostatic interactions between C60 molecules andgold electrodes. The observed current-voltage curves areshown in Fig. 18. The device fabricated by Park et al. isan example of a molecular electronic device [93] in whichelectrical conduction occurs through single moleculesconnected to conventional leads. The junctions betweenmolecular components and leads will be much more flexi-ble than those in conventional solid-state nanostructuresand fluctuations in their width may modify their cur-

FIG. 18: Current-voltage (I − V ) curves obtained from asingle-C60 transistor at T = 1.5 K. Five I − V curves takenat different gate voltages (Vg) are shown. Single-C60 transis-tors were prepared by first depositing a dilute toluene solutionof C60 onto a pair of connected gold electrodes. A gap of,1nm was then created using electromigration-induced break-ing of the electrodes. Upper inset, a large bias was appliedbetween the electrodes while the current through the con-nected electrode was monitored (black solid curve). After theinitial rapid decrease (solid arrow), the conductance stayedabove,∼ 0.05 mS up to ∼ 2.0 V. This behavior was observedin most single-C60 transistors, but it was not observed whenno C60 solution was deposited (red dotted curve). The biasvoltage was increased until the conductance fell low enoughto ensure that the current through the junction was in thetunneling regime (open arrow). The low bias measurementsshown in the main panel were taken after the breaking pro-cedure. Lower inset, an idealized diagram of a single C60-transistor formed by this method. From Ref. [6], H. Park etal., Nature, 407, 57 (2000), with permission from the Macmil-lan Publishers Limited c© 2000.

rent characteristics significantly. Furthermore, vibra-tional modes of the molecular components themselvesmay play an important role in determining the transportproperties [94].

An interesting possibility of a nanomechanical doublebarrier tunnelling structure involving shuttling has beenrealized by Majima et al. [95] and Nagano et al. [96]. Thesystem consists of scanning vibrating probe/colloidal Auparticles/vacuum/PtPd substrate, see Fig. 19. The col-loidal particles act as Coulomb islands, due to probe vi-bration they are brought to motion. What is importantis the phase shift between the probe vibrations and theAC current in the system which allows to single out thedisplacement current. The latter shows clear features ofthe Coulomb blockade. Comparison of the experimen-tal results with theoretical calculation drove the authorsto the conclusion that about 280 Au particles vibrate inaccordance with each other. The existence of such vibra-tions is also supported by experiments [97, 98] where anegative differential resistance of a shuttle structure wasobserved.

20

An externally driven nanomechanical shuttle has beendesigned by Erbe et al. [37, 38]. In these experimentsa nanomechanical pendulum was fabricated on Si-on-insulator substrate using electron and optical lithogra-phy, and a metallic island was placed on the clapperwhich could vibrate between source and drain electrodes,see Fig. 20. The pendulum was excited by applying anAC voltage between two gates on the left- and right-handsides of the clapper. The observed tunnelling source-drain current was strongly dependent on the frequencyof the exciting signal having pronounced maxima at theeigenfrequencies of the mechanical modes. This fact sig-nalizes a shuttling mechanism of electron transfer at typ-ical shuttle frequencies of about 100 MHz. The measuredaverage DC current at 4.2 K corresponded to 0.11±0.001electrons per cycle of mechanical motion. Both a theoret-ical analysis [50] and numerical simulations showed thata large portion of the voltage also acts on the island.The authors of Refs. [37, 38] expect that the resolution

FIG. 19: (a) The experimental arrangement of the tunnel-ing current and displacement current simultaneous measuringsystem. (b) A schematic image of nanomechanical double bar-rier tunneling structures with vibrating probe (W)/colloidalAu particles (diameter 8 nm)/vacuum/PtPd substrates. (c)The equivalent circuit of a two-junction system. C1 and R1

mean capacitance and tunneling resistance between a Au par-ticle and a PtPd substrate, and C2 and R2 are those betweena Au particle and a tungsten probe, respectively. C0 is thecapacitance between the tungsten probe and PtPd substrate.From Ref. [96], K. Nagano et al., Appl. Phys. Lett. 81,544 (2002), with permission from the American Institute ofPhysics c© 2002.

FIG. 20: Electron micrograph of the quantum bell: The pen-dulum is clamped on the upper side of the structure. It canbe set into motion by an ac-power, which is applied to thegates on the left and right hand side (G1 and G2) of the clap-per (C). Electron transport is then observed from source (S)to drain (D) through the island on top of the clapper. The is-land is electrically isolated from the rest of the clapper whichis grounded. From Ref. [37], A. Erbe et al.,Appl. Phys. Lett.73, 3751 (1998), with permission from the American Instituteof Physics c© 2001.

of the transport through the shuttle can be improved toalso resolve Coulomb blockade effects by minimizing theparasitic effects of the driving AC voltage. According totheir estimates, a Coulomb blockade should be observ-able below 600 mK. A very important modification ofthe setup in Fig. 20 was recently presented by Scheibleet al. in Ref. [39]. There a silicon cantilever is part of amechanical system of coupled resonators - a constructionthat makes it possible to drive the shuttle mechanicallywith a minimal destructive influence from the actuationdynamics on the shuttle itself. This is achieved by aclever design that minimizes the electrical coupling be-tween the driving part of the device (either a megneto-motively driven, doubly clamped beam resonator, or acapacitively coupled remote cantilever) and the drivenpart (the cantilever which carries the shuttle on its tip).Systems of the above mentioned type can, in principle,be used for studies of shuttle transport through super-conducting and magnetic systems. Further attempt todecrease the attenuation in a shuttle system was made in[99].

Interesting results on mechanically-assisted chargetransfer were obtained by Scheible et al. [58] in a device isfabricated as a silicon nanopillar located between sourceand drain contacts, see Fig. 21. The device is manufac-tured in a two-step process: First, nanoscale lithographyusing a scanning electron microscope (SEM), and, sec-ond, dry etching in a fluorine reactive ion etcher (RIE).The lithographically defined gold structure acts as bothelectrical current leads and etch mask for the RIE. Asimple geometry defined by the SEM consequently re-sults in the free-standing isolating nanopillar of intrinsic

21

FIG. 21: (a) SEM micrograph and experimental circuitry ofthe silicon nanopillar: At source S, an AC signal, Vac, isapplied with a superimposed DC bias V . The net current,ID is detected at drain D with a current amplifier. The thirdelectrode, G, is floating. (b) Finite element simulation ofthe base oscillation mode which compiles for the nanopillarto f0 = 5367 MHz. (c) When the island is deflected towardone electrode, the instantaneous voltage bias determines thepreferred tunneling direction. Cotunneling is absent in thiscase, due to an increased distance to the opposite electrode.From Ref. [58], D. V. Scheible and R. H. Blick, Appl. Phys.Lett. 84, 4632 (2004), with permission from the AmericanInstitute of Physics c© 2004.

silicon with a conducting metal (Au) island at its top, seeFig. 21,a. This island serves as the charge shuttle. Themetal island and the nanopillar are placed in the centerof two facing electrodes, denoted by source S and drainD. The system is biased by an AC voltage at source,rather than a sole DC bias, to avoid the dc-self excita-tion. Moreover, application of an ac-signal allows excita-tion of the nanopillar resonantly in one of its eigenmodes.The device itself is mounted in a probe station, provid-ing vacuum condition for a reproducible and controlledenvironment of the pillar. This also removes water andsolvents which may have condensed at the surface of theNEMS. The devices operated at room temperature andthe capacitance was not sufficiently small to realize thesingle-electron regime.

Experimentally, dependencies of the current throughthe system on bias frequency, as well as on additional DCbias allowing to tune resonances, were measured. The re-sults are qualitatively explained on the basis of numericalsimulations.

Coupling of electron transport to mechanical degreesof freedom can lead to other interesting phenomena. Inparticular, Kubatkin et al. [100] have observed a current-induced Jahn-Teller deformation of a Bi nanocluster.They have shown that such a transformation influences

the electron transport through a change in the geometri-cal shape of the cluster. We do not review here in detailother experiments, e.g., [6–12] involving nanoelectrome-chanical phenomena since they are not directly connectedto shuttle charge transport.

4. COHERENT TRANSFER OF COOPERPAIRS BY A MOVABLE GRAIN

In this Section we study a superconducting weak linkwhere the coupling between two bulk superconductorsis due to Cooper pairs tunnelling through a small mov-able superconducting grain. The system is depicted inFig. 22. We begin by looking at the requirements one hasto put on the system for the analysis below to be valid.Then we briefly review the Parity effect and the single-Cooper-pair box in Sec. 4.2. Following this, in Sec. 4.3,we consider the two basic processes involved in shuttlingof Cooper pairs; (i) scattering of a single grain with alead, thereby creating entanglement; and (ii) free motionof a grain whose charge state is a quantum mechanicalsuperposition.

Two possible experimental configurations can be imag-ined for the system in Fig. 22

1. The pair of remote superconductors might be cou-pled by an external superconducting circuit, form-ing a loop. In this case the superconducting phasedifference is given (and is for example determinedby a magnetic flux through the loop). Shuttlingof Cooper pairs is in this case a mechanism allow-ing for supercurrent flow through the loop. Thisscenario is analyzed in Sec. 4.4.

2. A qualitatively different situation is when the twoleads are disconnected from each other. Cooperpair exchange between two remote and isolated su-perconductors (leads) is then allowed only throughtunnelling via the single Cooper pair box, perform-ing oscillatory motion between leads. In this casethe relevant question is whether or not phase co-herence between the leads can be established. Thissituation is considered in Sec. 4.5.

4.1. Requirements for shuttling of Cooper pairs

The main question we will focus on is how mechani-cal vibration of the cluster affects coherent tunnelling ofCooper pairs. To put the question in a more dramaticway: Could one have coupling between remote supercon-ductors mediated mechanically through inter-lead trans-portation of Cooper pairs, performed by a small movablesuperconductor? The analysis given here, leading to apositive answer to this question has in parts been pre-sented in Refs. [101] and [102]. This result follows fromthe possibility to preserve phase coherence of Cooper

22

L R

Gates

Superconducting leads

Movable super-conducting grain

FIG. 22: Superconducting shuttle junction. Two supercon-ducting leads are placed too far away from each other to al-low for direct tunneling between them. Using a movable grainthey may still exchange Cooper pairs indirectly via lead-graintunneling. If the gates are appropriately biased a single-Cooper-pair box situation occurs close to either lead allow-ing for a coherent superposition of two different charge stateson the grain. Hence, for a grain making repeated alternatecontacts with the leads a coherent exchange of Cooper pairsbetween them is possible. From Ref. [40], R. I. Shekhter etal., J. Phys: Condens. Matter 15, R441 (2003), with permis-sion from the Institute of Physics and IOP Publishing Limitedc© 2003.

pairs despite the non-stationary and non-equilibrium dy-namics of the electronic system originating from a timedependent displacement of the small superconductingmediator. It is necessary though, that only a few elec-tronic degrees of freedom are involved in the quantumdynamics. The latter criterion is guaranteed to be ful-filled if two conditions are fulfilled:

1. The energy quantum ~ω0 associated with the vi-brations is much smaller than the superconductinggap ∆ .

2. The charging energy EC is much larger than the su-perconducting coupling energy EJ and the thermalenergy kBT .

Here EJ is the maximal Josephson energy characterizingthe superconducting coupling between grain and leads.Condition (i) prevents the grain motion from creating el-ementary electronic excitations and therefore guaranteesthat the quantum evolution of the system is disconnectedfrom any influence of the continuous spectrum of quasi-particles in the superconductors. Condition (ii) guaran-tees a Coulomb blockade for Cooper pair tunnelling andhence prevents significant charge fluctuations on the dot.Such fluctuations imply the existence of a large numberof channels for Cooper pair transport and result in strongdecoherence due to destructive interference between thedifferent channels. In what follows we will consider theconditions (i) and (ii) to be fulfilled.

4.2. Parity effect and the Single-Cooper-pair box

The ground state energy of a superconducting grain de-pends in an essential way on the number N of electronson it. Two important N-dependent contributions are theelectrostatic Coulomb energy EC(N), connected with theextra charge accumulated on the superconducting grain,and the so called parity term ∆N [103–105]. The latteroriginates from the fact that only an even number of elec-trons can form the BCS ground state of a superconductor(which is a condensate of paired electrons). Therefore, ifthe number of electrons N is odd, one unpaired elec-tron has to occupy one of the quasiparticles states. Theenergy cost for occupying a quasiparticle state, which isequal to the superconducting gap ∆, brings a new energyscale to bear on how many electrons can be accomodatedby a small superconducting grain. In view of this discus-sion the ground state energy E0(N) can be expressed as(see [105])

E0(N) = EC(N − αVg)2 +

0 even N∆ odd N .

(12)

One can see from (12) that if ∆ > EC only an evennumbers of electrons can be accumulated in the groundstate of the superconducting grain. Moreover, for specialvalues of the gate voltage Vg, such that αVg = 2N + 1,the ground state is degenerate with respect to a changeof the total number of electrons by one single Cooperpair. An energy diagram illustrating this case is pre-sented in Fig. 23. The occurrence of such a degeneracybrings about an important possibility to create a quan-tum hybrid state at low temperatures, which will be acoherent mixture of two ground states, differing by a sin-gle Cooper pair:

|Ψ〉 = γ1 |n〉+ γ2 |n + 1〉 . (13)

This coherent superposition state has been realized ex-perimentally in so called single-Cooper-pair boxes [106].Coherent control of the state (13) was first demonstratedby Nakamura et al. [107] and then confirmed and im-proved by Devoret et al. [108] and others. The idea ofthe experiment is presented in Fig. 24 where the super-conducting dot is shown to be in tunnelling contact witha bulk superconductor. A gate electrode is responsible forlifting the Coulomb blockade of Cooper pair tunnelling,thereby creating the ground state degeneracy discussedabove. This allows for delocalization of a Single Cooperpair between two superconductors. Such a hybridizationresults in a certain charge transfer between the bulk su-perconductor and the grain.

4.3. Basic principles

In order to realize the idea of Cooper pair shuttlinga rather straightforward extension of the single-Cooper-pair box experiments mentioned in the previous Section

23

FIG. 23: Energy diagram for the ground state of a supercon-ducting grain with respect to charge for the case ∆ > EC .For a certain bias voltage αVg = 2n + 1 [see Eq. (12)] groundstates differing by only one single Cooper pair becomes degen-erate. From Ref. [40], R. I. Shekhter et al., J. Phys: Condens.Matter 15, R441 (2003), with permission from the Instituteof Physics and IOP Publishing Limited c© 2003.

is required. Essentially, to shuttle Cooper pairs, one usesthe hybridization for coherent loading (unloading) of elec-tric charge to (from) a movable single-Cooper-pair boxthat transports the loaded charge between the remotesuperconductors. The necessary setup should contain amovable Cooper-pair box capable of performing forcedoscillations between two gated superconducting leads asshown in Fig. 22. The two gate electrodes ensure the lift-ing of the Coulomb blockade of Cooper-pair tunnelling atthe turning points in the vicinity of each of the supercon-ducting electrodes.

4.3.1. Scattering and free motion

To illustrate the shuttling process, consider first thesimple case when an initially uncharged grain, n = 0,gets into contact with the left lead. Before contact thestate of the system is

|Ψ(t0)〉 = |0〉 ⊗ |ψLeads〉 ,

where |ψLeads〉 is the state of the leads. During the timespent in tunnelling contact with the lead the Cooper pairnumber on the grain may change. When the grain ceasesto have contact with the lead the general state of thesystem is therefore

|Ψ(t1)〉 = α |0〉 ⊗ |ψ′Leads(t1)〉+ β |1〉 ⊗ |ψ′′Leads(t1)〉 .

This process is depicted in Fig. 25. The coefficients αand β are complex numbers and will depend both on thetime spent in contact with the lead and on the initial

FIG. 24: A schematic diagram of a single Cooper pair box.An island of superconducting material is connected to a largersuperconducting lead via a weak link. This allows for coher-ent tunneling of Cooper pairs between them. For a nanoscalesystem, such quantum fluctuations of the charge on the islandare generally suppressed due to the strong charging energy as-sociated with a small grain capacitance. However, by appro-priate biasing of the gate electrode it is possible to make thetwo states |n〉 and |n + 1〉, differing by one Cooper pair, havethe same energy (degeneracy of ground state). This allows forthe creation of a hybrid state |Ψ〉 = γ1 |n〉+ γ2 |n + 1〉. FromRef. [40], R. I. Shekhter et al., J. Phys: Condens. Matter 15,R441 (2003), with permission from the Institute of Physicsand IOP Publishing Limited c© 2003.

FIG. 25: A grain initially carrying zero excess Cooper pairsscatters with the left lead. After the scattering the state ofthe grain will be a superposition of zero and one extra Cooperpairs. In the process, the grain becomes entangled with theleads. From Ref. [40], R. I. Shekhter et al., J. Phys: Condens.Matter 15, R441 (2003), with permission from the Instituteof Physics and IOP Publishing Limited c© 2003.

state |ψLeads〉. We note here that in general the grainwill become entangled with the leads.

As the grain traverses the region between the leadsthere is no tunnelling and the magnitudes |α| and |β| willremain constant. However, the relative phase betweenthem may change. Thus, when the grain arrives at theright lead at a time t2 the state of the system will haveacquired an additional phase labeled χ+

|Ψ(t2)〉 = α |0〉 ⊗ |ψ′Leads(t2)〉+ e−iχ+β |1〉 ⊗ |ψ′′Leads(t2)〉 .

24

As the grain comes into contact with the right lead chargeexchange is again possible and the coefficients α and βwill change. Then, in the same fashion as during themotion from left to right, the only effect on the stateas the grain moves towards the left lead again is thatanother relative phase denoted by χ− is acquired. Thewhole process is schematically illustrated in Fig. 26.

Contact region Transportation region Contact region

Grain position

energy

Accumulation ofrelative phases c

+-

EJ

LE

J

R

EC

Josephsontunneling

Josephsontunneling

L R

t

t t1

0

2

t4

FIG. 26: Illustration of the charge transport process. The is-land moves periodically between the leads. Close to each turn-ing point lead-grain tunneling takes place. At these points thestates with net charge 0 and 2e are degenerate. As the grainis moved out of contact the tunneling is exponentially sup-pressed while the degeneracy may be lifted. This leads to theaccumulation of electrostatic phases χ±. From Ref. [40], R. I.Shekhter et al., J. Phys: Condens. Matter 15, R441 (2003),with permission from the Institute of Physics and IOP Pub-lishing Limited c© 2003.

Both the ”scattering events” and the ”free motion”are thus characterized by quantum phases accumulatedby the system. Here we refer to them as the Josephsonphase, ϑ, and the electrostatic phase, χ±:

ϑ = ~−1

∫dtEJ(t) , (14)

χ± = (i/~)∫

dt δEc[x(t)] . (15)

4.3.2. Hamiltonian

Under the condition that the variation of the grain po-sition x is adiabatically slow as discussed above, no quasi-particle degrees of freedom are involved and one only hasto consider the quantum dynamics of the coupled groundstates on each of the superconductors. The correspond-ing Hamiltonian is expressed in terms of the Cooper-pairnumber operator n for the grain and the phase operatorsfor the leads, ΦL,R,

H(x(t)) = −12

∑

s=L,R

EsJ(x(t))

(eiΦs |1〉〈0|+ h.c.

)

+δEC(x(t))n . (16)

The operator |0〉 〈1| changes the charge on the grain fromzero to one extra Cooper-pair. The Hamiltonian (16) rep-resents a standard approach to the description of super-conducting weak links [109]. However an essential specificfeature here is the dependence of the charging energy dif-ference δEC and the coupling energies EJ on the positionof the superconducting grain x,

EL,RJ (x) = E0 exp (−δxL,R/λ) .

Here δxL,R is the distance between the grain and therespective lead. Since the displacement x is a given func-tion of time we are dealing with a non-stationary quan-tum problem.

4.4. Transferring Cooper pairs between coupledleads

When the leads are completely connected, i.e. they aresimply different parts of the same superconductor, thereis, due to charge conservation, a one-to-one correspon-dence between the number of pairs on the conductor andthe number of pairs n on the island. One may in this caseassume the leads to be in states with definite phases,

|ψLeads〉 = |ΦL〉 ⊗ |ΦR〉 .

The effect of this is to replace the operators e±iΦL,R inthe Hamiltonian above with c-numbers making it a two-level system. This leaves us with a reduced Hamiltonianwhere the phases ΦL,R enter only as parameters,

Hred(x(t),ΦL, ΦR) ≡ 〈ΦL| 〈ΦR|H(x(t)) |ΦR〉 |ΦL〉 .(17)

The dynamics of the system is described by theLiouville-von Neumann equation for the density matrixρ [101],

dρ/dt = −i~−1[Hred, ρ]− ν(t)[ρ− ρ0(t)] . (18)

Since we are not interested here in transient processes,connected with the initial switching on of the mechani-cal motion, solutions which do not depend on any initialconditions will be in the focus of our analysis. To preventany memory of initial conditions one needs to include adissipation term (the last term on the right hand side ofEq. (18)) into the dynamics. If this dissipation is weakenough it does not affect the system dynamics on a timescale comparable to the period of vibrations. Howeverin a time scale longer than the period of rotation suchrelaxation causes solutions to Eq. (18) to be independentof any initial conditions. We use the simplest possible re-laxation time approximation (τ -approximation) with ρ0

being an equilibrium density matrix for the system de-scribed by Hamiltonian H. Relaxation is due to quasi-particle exchange with the leads and depends on the tun-nelling transparencies

ν(x) = ν0 exp(−δx/λ) . (19)

25

The Cooper-pair exchange, being an exponential functionof the grain position, is mainly localized in the vicinityof the turning points. In this region the Coulomb block-ade of Cooper-pair tunnelling is suppressed and can beneglected while considering the dynamics of the forma-tion (or transformation) of the single-Cooper-pair hybrid(13). In contrast, the dynamics of the system during thetime intervals when the grain is far away from the super-conducting leads, is not significantly affected by Cooper-pair tunnelling and essentially depends on the electro-static energy δEC = EC(n + 1) − EC(n) that appearsas the Coulomb degeneracy is lifted away from the turn-ing points. This circumstance allows us to simplify theanalysis and consider the quantum evolution of the sys-tem as a sequence of scattering events and “free motion”.Scattering takes place due to tunnelling of Cooper pairsin the vicinity of the turning points and these events arespeparated by intervals of free evolution of the system,where any tunnelling coupling between grain and leads isneglected. A schematic picture of the described sequenceis presented in Fig. 26.

Direct calculations [101] give a simple expression forthe average current,

I = 2efcos ϑ sin3 ϑ sinΦ (cos χ + cosΦ)1− (cos2 ϑ cosχ− sin2 ϑ cosΦ)2

, (20)

where Φ = (ΦR − ΦL) + (χ+ − χ−) and χ = χ+ + χ−(for details of the derivation see also [101, 110]). Thefollowing features of Eq.(20) should be mentioned:

1. The mechanically assisted supercurrent is an os-cillatory function of the phase differences of thesuperconducting leads similarly to other types ofsuperconducting weak links.

2. The current can be electrostatically controlled if anasymmetrical (χ+ is not equal to χ−) phase accu-mulation is provided by an external electric fieldvarying in tact with the grain rotation. The sameeffect appears if the grain trajectory embeds a finitemagnetic flux. Then χ is proportional to the fluxgiven in units of the superconducting flux quantum.

3. Depending on the value of the electrostatic phase χone can have any direction of the supercurrent flowat a given superconducting phase difference. Also anonzero supercurrent at ΦL−ΦR = 0 is possible incontrast to ordinary superconducting weak links.

4. The supercurrent is a non-monotonic function ofthe Josephson coupling strength ϑ. This fact re-flects the well known Rabi oscillations in the pop-ulation of quantum states with different numbersof Cooper-pairs when a single-Cooper-pair box isformed due to sudden switching of pairs tunnellingat the turning points of the trajectory.

In the weak coupling limit the current is proportionalto the third power of the maximal Josephson coupling

FIG. 27: The magnitude of the current I in Eq. (20) in unitsof I0 = 2ef as a function of the phases Φ and χ. Regionsof black correspond to no current and regions of white to|I/I0| = 0.5. The direction of the current is indicated in thefigure by signs ±. To best see the triangular structure of thecurrent the contact time has been chosen to ϑ = π/3. I0 con-tains only the fundamental frequency of the grain’s motionand the Cooper pair charge. From Ref. [40], R. I. Shekhter etal., J. Phys: Condens. Matter 15, R441 (2003), with permis-sion from the Institute of Physics and IOP Publishing Limitedc© 2003.

strength. One needs to stress that this strength mightbe several orders of magnitude bigger than what a directcoupling between the superconducting leads would give.Cooper-pair transportation serves as an alternative todirect Cooper-pair tunnelling between the leads therebyproviding a mechanism for supercurrent flow between theremote superconductors. In Fig. 27 a diagram of the thesupercurrent as a function of both superconducting andelectrostatic phases is presented.

4.5. Shuttling Cooper pairs between disconnectedleads

We now turn to the question of whether or not a super-conducting coupling between remote and isolated super-conductors can be mediated by mechanically shuttlingCooper-pairs between them [102]. Here we are interestedin the situation when up to a time t0 the shuttle is absentand no well defined superconducting phases can be intro-duced for the leads. This is because of the large quantumfluctuations of the phases on the conductors, which eachcontain a fixed number of Cooper-pairs. At times t largerthan t0, a superconducting grain starts to swing betweenthe leads, and the number of Cooper-pairs on each leadis no longer conserved separately. The moving single-Cooper-pair box provides a mechanism for exchange ofCooper-pairs between the leads. We are interested to

26

know if this exchange is able to establish superconduct-ing phase coherence between the leads.

In the case with strongly coupled leads the limitationthat the grain could carry only zero or one excess Cooperpair reduced the problem to a two state problem. Sincethe problem was essentially a two conductor problem(grain + one lead) only this variable was needed to de-termine the state of the system. For the case with decou-pled leads one has to keep the operator nature of ΦL,R inthe Hamiltonian (16). The dimensionality of the Hilbertspace depends on the maximum number of Cooper pairsthat can be accommodated on the leads. The factorsthat put a limit to this number are the capacitances ofthe leads which are not present in the Hamiltonian (16).Instead of including these charging energy terms anotherapproach was used in Ref. [102]. The Hilbert space wasthere reduced in such a way that each lead can only ac-commodate a maximum (minimum) of ±N extra Cooperpairs.

The time evolution of the system is determined by theLiouville-von Neumann equation for the density matrix.If the total number of particles in the system is conservedand one assumes the whole system to be charge neutral,the state of the system can be written in terms of thestate of the grain and one of the leads. The density ma-trix can, e.g., be written as

ρ(t) =∑

η,σ=0,1

N∑

NL,N ′L=−N

ρησNLN ′

L(t) (21)

× |η〉 〈σ|︸ ︷︷ ︸grain

⊗ |NL〉 〈N ′L|︸ ︷︷ ︸

left lead

⊗ |−NL − η〉 〈−N ′L − σ|︸ ︷︷ ︸

right lead

.

Here, it has been explicitly indicated to which part ofthe system the various operators belong. The simple re-laxation time approximation used for the case with con-nected leads is not possible to use in this case. That ap-proximation assumed the leads to be in BCS states withdefinite phases towards which the phase of the grain re-laxed. Instead, to account for loss of initial conditionsthe influence of the fluctuations of the gate potential onthe island charge has been accounted for. The fluctua-tions are modelled by a harmonic oscillator bath with aspectral density determined by the impedance of the gatecircuit. Hence, the density matrix in Eq. (21) is the re-duced density matrix obtained after tracing out the bathdegrees of freedom.

Denoting the phase difference between the leads by∆Φ ≡ ΦR−ΦL and the phase difference between the rightlead and the grain ∆φ ≡ ΦR − φgrain, phase differencestates

|∆Φ, ∆φ〉 ≡ 12π

∑n

∑

NL

e−iNL∆Φe−in∆φ

×|n〉|NL〉| − (NL + n)〉

are introduced. The probability for finding a certainphase difference between the leads is then obtained from

FIG. 28: Probability for finding a phase difference ∆Φ be-tween the leads as a function of the number of grain rota-tions. From Ref. [102], A. Isacsson et al., Phys. Rev. Lett.89, 277002 (2002), with permission from the American Phys-ical Society c© 2003.

the reduced distribution function

f(∆Φ) ≡2π∫

0

d(∆φ) 〈∆Φ, ∆φ| ρ |∆Φ,∆φ〉

=12π

∑

NL,N ′L

∑η

e−i(N ′L−NL)∆Φρηη

NLN ′L

.

This is the function which has been plotted on the z-axis in Fig. 28. At the initial stage of the simulation thesystem was in a state with a definite amount of charge ineach conductor. This means that the phase distributionis initially completely flat. However, as the grain rotatesthe distribution is altered and eventually becomes peakedaround a definite value of ∆Φ uniquely determined by thesystem parameters. The width of the final peak dependson the maximum number of excess Cooper pairs that canbe accommodated on the leads.

The phase difference mediated by shuttling Cooperpairs will give rise to a current if the two conductors areconnected by an ordinary weak junction. In Fig. 29 thiscurrent is shown as a function of the dynamical phasesfor fixed ϑ (contact time). Here an auxiliary, weak, probeJosephson junction is assumed to be connected after alarge number of rotations. The current is given by theusual Josephson relation weighted over the phase distri-bution f(∆Φ)

I = Ic

2π∫

0

d(∆Φ) sin(∆Φ)f(∆Φ),

where Ic is the critical current of the probe junction.We conclude that phase coherence can be established

by a mechanical transfer of Cooper pairs and that thismechanism can also give rise to a non dissipative current.

27

FIG. 29: Current through a probe Josephson junction con-nected between the leads after many rotations. Bright areascorrespond to large current while black ones correspond tozero current. The signs indicate the direction of the current.From Ref. [102], A. Isacsson et al., Phys. Rev. Lett. 89, 277002(2002), with permission from the American Physical Societyc© 2003.

The role of an environment-induced decoherence in theshuttling of Cooper pairs was considered by Romito etal. [111]. To allow for decoherence a finite model damp-ing of the off-diagonal part of the density matrix is in-troduced in the Liouville-von Neumann equation (18).The damping is assumed to be different in the con-tact regions (γJ) and in the region between the con-tacts (γC) and computed in the Born-Markov approx-imation. Strong decoherence exponentially suppressesthe supercurrent, the leading term being proportional toexp(−γJ t0 − γCtC), where tC = t − t0. The supercur-rent is also suppressed in the case of weak coupling andγJ t0 ¿ γCtC . Consequently, there exists an intermediateregion where the decoherence leads to an enhancement ofthe supercurrent. In addition, the decoherence may re-sult in a sign change (π junction). It remains to be seenwhether these conclusions are sensitive to damping of thediagonal elements of the density matrix.

5. NOISE IN SHUTTLE TRANSPORT

Noise properties are crucial for the performance ofnanomechanical systems and have been extensively stud-ied both experimentally and theoretically. In this reviewwe address only work related to shuttle electron trans-port.

While many papers were aiming at studies of the condi-tions for the realization of the shuttle instability or at thedependence of the current on external parameters, at theinitial stage only a few papers investigated current fluc-

tuations. However, during the last several years the in-terest in current fluctuations have significantly increased,and has been shown that the current noise contains muchmore information about the nature of the shuttle insta-bility then the average current. Indeed, even near equi-librium, the noise spectrum allows one to study the ACresponse of the system without AC excitation. Out ofequilibrium, the noise spectrum is specifically sensitiveto coherence properties of the electron transport, as wellas to electron-electron correlations. In addition, noisespectrum can serve as a hallmark of shuttle transportand can be used for its detection. The above issues leadto several interesting works aimed at various aspects ofnoise in electron shuttling. Below we will discuss theseworks following the scheme of Table I.

5.1. General concepts

Before we proceed, let us introduce some basic defini-tions. The instant current through a device, I(t), differsfrom its time-averaged value, I ≡ I(t), and the difference∆I(t) ≡ I(t) − I is called the current fluctuation. Thenature of fluctuations is naturally studied by evaluatingthe correlation function

∆I(t)∆I(t′) = I(t)I(t′)− I2 .

In the absence of external time-dependent fields this cor-relation function depends only on the time difference,t − t′. In addition, for the ergodic systems we are in-terested in, the time average does not differ from theensemble average, which we will denote as 〈· · · 〉. Theensemble average is just the average over the realizationsof the random quantity I(t). As a result, the correlationfunction is conventionally defined as

S(τ) ≡ 〈∆I(τ)∆I(0)〉 . (22)

The noise spectrum is then defined as twice the Fouriertransform of the correlation function, see Ref. [112] fora review,

S(ω) ≡ 2∫ ∞

−∞dτ e−iωτS(τ) . (23)

This is a purely classical definition, which assumes thatthe current operators commute at different times. In thecase of quantum transport the current is an operator,and in general I(t) and I(t′) do not commute. Then thedefinition of the correlation function is generalized as

S(τ) =12

⟨[I(τ), ˆI(0)

]+

⟩− 〈I〉2 , (24)

where [A, B]± ≡ AB ± BA. For a small applied voltage,V → 0, the current is proportional to the applied electricfield. This implies that I(ω) = G(ω)E(ω), where G(ω) isthe complex conductance of the structure. In this regime

28

the noise spectrum can be expressed through the realpart of the conductance as

S(ω) = ~ω Re G(ω) coth~ω

2kBT≈ 2kBT Re G(ω) , (25)

where the approximate result holds if ~ω ¿ kBT ; kB isthe Boltzmann constant. The relation (25) is the wellknow fluctuation-dissipation theorem [113]. In the linearresponse regime it follows from Eq. (25) that the noisespectrum provides exactly the same information as thelinear conductance. Even in this case studies of noisecan be informative since the noise spectrum allows one todetermine the frequency dependence of the conductancewithout a direct AC excitation.

Nonequilibrium noise (V 6= 0) is more interesting, be-cause it gives information about temporal correlations ofthe electrons, that can not be obtained from the con-ductance. The contribution proportional to the firstpower of the applied bias voltage is often called the shotnoise. Such noise has been thoroughly studied in manysystems. In devices such as tunnel junctions, Schottkybarrier diodes, p − n junctions, and thermionic vacuumdiodes [114], the electrons are transmitted randomly andindependently of each other. Hence, the transfer of elec-trons can be described by Poisson statistics, which is usedto analyze events that are uncorrelated in time. For thesedevices the shot noise has its maximum value at zero fre-quency,

S(0) = 2e 〈I〉 ≡ SPoisson , (26)

and is proportional to the time-averaged current 〈I〉.This expression is valid for ω ¿ τ−1, where τ is the effec-tive width of a one-electron current pulse, which is deter-mined by the device parameters. For higher frequenciesthe shot noise vanishes. To characterize the shot noisethe so-called Fano factor, F ≡ S(0)/2e 〈I〉 is introduced.Correlations suppress the low-frequency shot noise belowthe Poisson limit, leading to F ≤ 1. This suppressionis efficiently used to study correlations in the electrontransport through mesoscopic devices, see the excellentreview articles Refs. [112, 115].

Even more interesting information can be extracted byusing so-called full counting statistics [116], which dealswith the probability distribution, Pt(n), of the number ofelectrons, n, transferred through the system during themeasurement time, t. The first and the second momentsof this distribution correspond to the average current andthe shot-noise correlations, respectively. The probabilitydistribution also contains fundamental information aboutlarge current fluctuations in the system.

In the following Section we review work on the noisespectrum and on the full counting statistics of shuttletransport.

5.2. Incoherent electron transport and classicalmechanical motion

The case of incoherent electron shuttling involvingclassical mechanical motion was first addressed by Isac-sson and Nord [117]. They considered a model one-dimensional shuttle structure similar to the one shownin Fig. 1,a and described in detail in Ref. [118]. Accord-ingly, it was assumed that a metallic grain of mass Mand radius r is suspended between two leads by elastic,insulating springs. Applying a bias voltage V = VL−VR,electron transport occurs by sequential, incoherent, tun-nelling between the leads and the grain. The tunnellingrates, Γ±L,R(x, q), depend on grain position x and chargeq through the tunnelling matrix elements and the differ-ences in (Gibbs) free energy, ∆G±L,R(x, q), between thecharge configurations q, qL,R and q ± e, qL,R ∓ e.The electric potentials and charges are determined byusing a conventional electric circuit, where the voltagesources VL and VR are connected in series with theleads and the grain as shown in Fig. 30. As a re-

FIG. 30:

sult, the quantities ∆G±L,R(x, q) are expressed throughposition-dependent capacitances specified as CL,R(x) =C

(0)L,R/(1± x/aL,R), where aL,R are characteristic length

scales. The tunnelling matrix elements are expressedthrough the position-dependent resistances specified asRL,R(x) = R

(0)L,Re±x/λ. In this way the motion-induced

feedback to the stochastic electron hopping is taken intoaccount. Another relationship between the grain chargeand displacement is given by Newton’s equation of mo-tion, mx = F (x, x, q). The force F in this equation con-tains both elastic and electric components, as well as thefriction force ∝ x. A new feature is an additional accountof the van der Waals force, which turns out to be impor-tant [118]. In general, F (x, x, q) is a nonlinear functionof x and q.

The results for noise were obtained by direct numeri-cal integration of a set of equations including the Masterequation for the grain population with x, q-dependenthopping rates and anharmonic mechanical equation ofmotion. A typical noise spectrum is shown in Fig. 31.The spectrum in this figure can be divided in four re-gions marked as I-IV. At the high frequencies of regionIV the Fano factor is close to 1/2, which is the value oneobtains for a static double junction [112]. In region III

29

108

1010

1012

0

0.5

1

1.5

2

2.5

3

Frequency [Hz]

I II III IV

FIG. 31: Power spectrum S(ω) in the shuttle regime. Forfrequencies above the vibrational frequency, the Fano factoris close to 1/2 as for a static Coulomb-blockade junction. Thepeaks are located at the frequency of vibration and at the firstharmonic. For frequencies below the vibrational frequency,the temporal correlation due to the periodic grain motionleads to a slight suppression of the noise level. At still lowerfrequencies, the noise is increased due to slow fluctuations inmechanical energy. From Ref. [117], A. Isacsson and T. Nord,E urophys. Lett. 66, 708 (2004), with permission from theEDP Sciences c© 2004.

two strong peaks are located at the vibration frequencyand its harmonic. This is a result of the periodic charg-ing and decharging of the oscillating grain. Directly be-low the peaks, in region II, the noise is suppressed belowthe shot noise level of a static double junction, due tothe additional time-correlations between successive tun-nel events induced by the oscillating grain. This is a clearhallmark of classical shuttling.

The most interesting part of the spectrum, however, isthe low-frequency part in region I, where the Fano factorincreases as the frequency decreases. The authors at-tribute this increase to low-frequency fluctuations in themechanical energy, which, in turn, lead to low-frequencyfluctuations in the current.

The authors have also performed an analytical stabilityanalysis valid in the case of weak electromechanical cou-pling, i.e. for ε ≡ F (x = 0, q = e)/mω2

0λ ¿ 1. The in-stability increment, p(E) = W(E)−D(E), is determinedby the difference between the the energy, W(E), pumpedinto the mechanical motion during one period and the av-erage energy dissipated per period, D(E). Hence, the sta-tionary oscillation amplitude is determined by the equa-tion p(E0) = 0. Since p(E) depends both on the biasvoltage and the damping of the mechanical motion, thisequation actually determines the dependence of the os-cillation amplitude on the bias voltage. It has a finitesolution only above the instability threshold.

The analysis, following the conventional proce-

dure [119], takes into account the fluctuations in themechanical energy around, say E = E0. These fluctua-tions induce electrical noise with a Lorentzian spectrum,S(ω) ∝ (ν2 + ω2)−1. The width ν of the spectrum isproportional to |p′(E0)|, where p′(E0) ≡ ∂p/∂E |E=E0 .Since at the instability threshold p′(E0) → 0 [42], thenoise spectrum diverges while approaching the instabil-ity threshold from the “shuttling” side.

The results of numerical studies corresponding to ananometer-sized Au grain commonly used in experimentswith self-assembled Coulomb-blockade double junctionsare shown in Fig. 32. Although, as explained in

FIG. 32: Current-voltage characteristics plotted togetherwith the Fano factor characterizing the noise spectrum in thestatic limit (ω → 0). The current is the solid line with thescale on the left ordinate while the Fano factor is shown fora discrete set of points with the scale on the right ordinate(lines connecting the points are a guide to the eye). Belowthe critical voltage where there is no sustained grain motion,the Fano factor is that of a Coulomb-blockade double junc-tion. Above the critical voltage, the grain is oscillating andthe Fano factor is increased and shows a divergent behaviorat the critical voltage. From Ref. [117], A. Isacsson and T.Nord, Europhys. Lett. 66, 708 (2004), with permission fromthe EDP Sciences c© 2004.

Ref. [118], the non-parabolic confining potential smearsany step-structure in the current-voltage characteristics,the transition between static- and shuttle-operation isclearly visible in the noise spectrum. In accordance withthe analytical result, on approaching the threshold fromabove (going from higher to lower voltages) the noisespectrum S(ω) shows a divergent behavior. Below thethreshold voltage the Fano factor is of the order 1/2.

A rather unusual prediction is that in the shuttleregime, well above the threshold voltage, the Fano factoris increased. This fact is attributed to the anharmonicityof the potential. For the harmonic potential used in theanalytical treatment the lowered noise level in region IIis continued into region I.

30

5.3. Noise in a quantum shuttle

We start with review of a theory by Novotny et al. [120]for shot noise in a quantum shuttle. The theory extendsprevious work of the authors [78] in which they consid-ered the average current. It is assumed that the shut-tling grain has two electron charge states, |0〉, and |1〉,and that only the diagonal elements of the density ma-trix in the |i〉-representation are important. To calculatethe noise, the number-resolved diagonal density matrices,ρ(n)ii (t) δik, are introduced. Here n is the number of elec-

trons which have tunnelled to the right electrode by timet. These matrices obey a generalized master equationwhere tunnelling into the leads is described by position-dependent transition rates ΓL,R. In addition, damping ofthe oscillator motion due to interacting with a thermalbath is taken into account.

Knowing ρ(n)ii (t), one finds the probability for n elec-

trons to be shuttled as

Pn(t) = Trosc∑

i=0,1

ρ(n)ii (t) .

The calculation of the average current and noise is thenstraightforward:

I = e limt→∞

∑n

nPn(t) ,

S(0) = 2e2 limt→∞

ddt

∑

n

n2Pn(t)−(∑

n

nPn(t)

)2 .

The relevant elements of the density matrix were foundusing the generating functional concept, and both the av-erage current and the Fano factor were calculated for dif-ferent relationships between the characteristic tunnellinglength, λ, and the amplitude of quantum zero-point os-cillations, x0, as well as for different ratios γ/ω0. The nu-merical results agree with an analytical treatment validfor small injection rates. The results are summarized inFig. 33.

The plot of I versus γ in Fig. 33 shows a crossoverfrom tunnelling to shuttling as damping is decreased, inagreement with previous results. The crossover spans anarrower range of γ-values in the case of λ/x0 = 2 com-pared to the λ/x0 = 1 case. Thus, already for λ/x0 = 2the shuttle behaves almost semiclassically, where a rel-atively sharp transition between the two regimes is ex-pected. There is no abrupt transition from tunnellingto shuttling, however, and near the transition theseregimes can coexist. To demonstrate this phenomenonthe Wigner distribution functions (7) were calculated fol-lowing Ref. [78].

The results are shown in Fig. 34, where a pure classicalmotion would correspond to a sharp classical phase tra-jectory for Wtot, which for an oscillator is a circle. The ra-dial smearing of the circle that can be seen corresponds toquantum fluctuations. In addition, one clearly sees a spot

FIG. 33: Current I (upper panel) and Fano factor F (lowerpanel; log scale) versus damping γ for different transfer ratesΓ = ΓL = ΓR and tunneling lengths λ. γ and Γ are measuredin units of ω0, λ is measured in units of x0. The corrent sat-urates in the shuttling (low damping) regime to one electronper cycle independently of the parameters while is substan-tially proportional to the electron transfer rate Γ = ΓL = ΓR

in the vibronic regime (high damping). The very low noise inthe shuttling (low damping) regime is a sign of ordered trans-port. The huge super-Poissonian Fano factors correspond tothe onset of the mixed regime. From Ref. [121], A. Donariniet al., New Journ. Phys. 7 237 (2005), with permission fromthe Institute of Physics and IOP Publishing Limited c© 2005.

in the center, which correspond to tunnelling through astatic grain. Thus the motion has a complex charactershowing features corresponding to both the classical andthe quantum regimes. The semiclassical transition is ac-companied by the nearly singular behavior of the Fanofactor reaching the value ≈ 600 at the peak. This is inagreement with the classical study [117] discussed above.

Noise spectum of a shuttle was analyzed in detail insubsequent works [121–123]. The general scenario of theevolution of noise spectra with increase of the bias isthe crossover between a tail with a maximum at zerofrequency to set of peaks in the shuttling regime – atslightly renormalized shuttling frequency, and its har-monics. Shown in Fig. 35 is an example of the noise

31

FIG. 34: Phase space picture of the shuttle around the transi-tion where the shuttling and tunneling regimes coexist. Fromleft to right we show the Wigner distribution functions for thedischarged, W00, charged W11, and both, Wtot =

∑i=0,1 Wii,

states of the oscillator in the phase space. From Ref. [120],T. Novotny et al., Phys. Rev. Lett. 92 248302 (2004), withpermission from the American Institute of Physics c© 2004.

spectra for different damping. Peaks in the noise spec-trum in the shuttling regime were also predicted forthe weakly coupled quantum electromechanical shuttlein Refs. [124, 125].

FIG. 35: The ratio between the current noise and the currentF (ω) = S(ω)/ 〈I〉 as function of the damping γ and frequencyω. The other parameters are ΓL = ΓR = 0.05ω0, λ = x0, d ≡eE/mω2

0 = 0.5x0, where x0 =√~/mω0. Peaks are seen at

ω ' 1.03ω0, 2.06ω0, 3.09ω0. The peak at ω ' 2.06ω0 reachesvalues F (ω) ' 20 (not shown) for γ = 0.02 and decreasesmonotonously with increasing γ to F (ω) ' 6 for γ = 0.09.The insert shows a representative curve (γ = 0.05). FromRef. [121], A. Donarini et al., New Journ. Phys. 7, 248302(2004), with permission from the Institute of Physics and IOPPublishing Limited c© 2005.

The peaks in the noise spectrum is a hallmark of shut-tling transport. Consequently, measurements of the noisespectra may be informative for identification of shuttling.In the mixed regime, a specific low-frequency dichotomicnoise originated from slow switching between the shut-tling was predicted and analyzed in [121]. Its presencemay identify the mixed regime.

An alternative method for studies of quantum noise inthe electromechanical shuttle was suggested in Ref. [126].The method is based on the solution and subsequent

analysis of a quantum master equation for the densitymatrix in the time domain. This method allows analyz-ing the conditional dynamics (system evolution statingfrom prescribed initial conditions) of the quantum shut-tle and identify specifics of the dynamics in real time.Applied to the current fluctuations, this approach alsopredicts a set of equidistant peaks in noise spectrum inthe shuttling regime. At relatively large damping theresult of the two above mentioned approaches basicallyagree. Some discrepancies at very small damping areprobably due to complications in numerics at very lowdamping.

5.4. Driven charge shuttle

The noise properties of a driven charge shuttle [38]are much simpler to account for than those of a self-oscillating shuttle. Here the mechanical energy is con-served and does not fluctuate, so the only source of noiseis random electron transfer.

The average number of electrons transferred by adriven shuttle per one cycle, as as well as its variancewere considered by Weiss and Zwerger [50] in connec-tion with a discussion of the accuracy of a mechani-cal single-electron shuttle, see Sec. 2.1.4. The variance,∆n ≡ 〈n2 − n2〉1/2 was found using a conventional Mas-ter equation approach with the tunnelling probabilitiesgiven by the ”orthodox theory”, see Ref. [127] for a re-view. A typical plot of ∆N versus V is shown in Fig. 5(lower panel), and the main conclusion is that the vari-ance is small at low temperatures, kBT ¿ e2/2C, andfor relatively long contact times, t0 À RC.

However, the variance ∆n differs from the noise actu-ally measured since typical measurement times are muchlonger than one period of oscillation. This case was ad-dressed by Pistolesi in Ref. [128] where both the zero-frequency noise and the full counting statistics of thetransferred charge were considered.

To find the statistics of the transferred charge oneneeds the probabilities for n electron to be transferred,Pn(t), for all n. They can be calculated using an elegantformalism involving a generating functional [116]. Thegenerating functional is defined as

e−Gt(χ) =∞∑

n=0

Pt(n) einχ (27)

where χ is the counting field. Then all cumulants of thetransferred charge can be calculated as

n = 〈n〉 = i∂G∂χ

∣∣∣∣χ=0

,⟨n2 − n2

⟩=

∂2G∂χ2

∣∣∣∣χ=0

, (28)

etc. The explicit calculation uses the method developedin Ref. [129] with a proper generalization to the dynamiccase. It is assumed that the shuttle has two states with 0or 1 excess electron. The probability to find the shuttle in

32

one of these states can be expressed as a vector |p〉 withcomponents p0, p1. The dynamics of |p(t)〉 is governedby the Liouville equation

∂

∂t|p(t)〉 = L |p(t)〉 , L =

(ΓL(t) −ΓR(t)

−ΓL(t) ΓR(t)

). (29)

The generating functional can be expressed as

e−Gt(χ) =⟨

q

∣∣∣∣T exp−

∫ t

0

Lχ(t′) dt′∣∣∣∣ p(0)

⟩, (30)

where |p(0)〉 is the probability at time t = 0, |q〉 ≡ 1, 1,and T exp is the time ordered exponential. The matrixLχ(t) is constructed from the matrix L(t) by multiplyingthe lower off-diagonal matrix element by the factor eiχ.Time ordering is very important since the matrices Lχ(t)do not commute for different times.

The generating functional (30) was analyzed numeri-cally as well as analytically for the limiting cases of smalland large oscillation amplitude a. In the static case,a = 0, and for a symmetric shuttle with ΓL = ΓR =Γ(0) exp(∓a sin ω0t) the known result [130] for a statictunnelling system is reproduced:

Gt(χ) = −Γ(0)t(eiχ/2 − 1

). (31)

In this case I = S(0) = πω0Γ(0) and the Fano factorF = 1/2.

In the opposite limit of large shuttle oscillation am-plitude the ratio ΓL/ΓR is most of the time either verylarge or very small. This is why it was assumed that (i)for 0 < ω0t < π the quantity ΓL vanishes identically and(ii) for π < ω0t < 2π the opposite holds: ΓR = 0. Theapproximation becomes exact for Γ ¿ ω0, since in thatcase electrons can tunnel only when the shuttle is nearone of the two leads. Under this assumption the problemcan be treated analytically, the result being

I = e1− α

1 + α, S(0) = 4e2α

1− α

(1 + α)3→ F =

2α

(1 + α)2.

Here the quantity 1− α with

α = exp(−Γ(0)

ω0

∫ π

0

dφ ea sin φ

)

is the probability of transferring one electron during halfa cycle. For a ¿ 1 this probability is nearly 1, and thegenerating function

e−Gt(χ) =[2α + (1− 2α) eiχ

]ω0t/2π(32)

corresponds to a binomial distribution

Pn(N) =(

Nn

)(1− 2α)n(2α)N−n,

where N(t) = [ω0t/2π] is the number of oscillation cy-cles during the measurement time t. This is a very clear

result since during each cycle one electron is transmit-ted with probability 1 − 2α, and since α ¿ 1 the cyclesare independent. Indeed, after each cycle the system isreset to the stationary state within accuracy α2, regard-less of the initial state. This limiting case agrees withthe results of Ref. [50], where the variance of the chargetransfer during one cycle was analyzed.

For α → 1 the probability for one electron to tunnelduring a cycle is very small. The result for this case readsas

e−Gt(χ) =[α + (1− α) eiχ/2

]ω0t/2π

(33)

One can notice that the periodicity of the generatingfunction has changed. Equation (33) describes a systemof e/2 charges that in each cycle are transmitted withprobability 1 − α). Thus the system can be mapped ona fictitious system of charges e/2 saying that every timethat one electron succeeds in jumping on or off the centralisland, one charge e/2 is transmitted in the fictitious sys-tem. This is possible, since it is extremely unlikely thatone electron can perform the full shuttling in one cycle.Thus after many cycles (N À 1) the counting statisticsof these two systems coincide. The cycles are no moreindependent as in the case when α ¿ 1, but the problemcan be mapped onto an independent tunnelling problem.For intermediate values of α it is more difficult to givea simple interpretation of the charge transfer statistics,since different cycles are correlated in a nontrivial way.

The full counting statistics of nanoelectromechanicalsystems was recently addressed by Flindt et al.[131]. Theauthors have developed a generalized theory applicableto a broad class of nanoelectromechanical systems whichcan be described by a generalized Markovian Masterequation. Concrete calculations are made for the modelsof Refs. [64] and [78, 79, 120]. The three first cumu-lants are evaluated numerically. For the quantum shut-tle [78, 79, 120], the behavior of the third cumulant isshown to be compatible with the concept of slow switch-ing between the tunnelling and the shuttling regime. Thisconcept was earlier [120] used to predict n enhanced noisespectrum at the shuttling transition.

Generally, both the noise and the full counting statis-tics demonstrate a very rich and interesting behavior.This permits us to understand more deeply the chargetransfer dynamics and to characterize the threshold forthe shuttle instability with greater accuracy.

5.5. Noise in Cooper pair shuttling

Noise in Cooper pair shuttling between two supercon-ductors is particularly interesting since in allows one tobetter understand the coherent properties of supercon-ductor devices. At present, only driven superconductingshuttle systems have been considered. Their noise prop-erties were first considered by Romito et al. [111].

33

The main purpose of Ref. [111] was to analyze envi-ronmentally induced decoherence, the noise being a by-product of a general analysis of the Cooper pair shut-tling dynamics. The basic model is similar to that ofRefs. [101, 102]. In addition, a finite coupling to a ther-mal bath was taken into account along the lines of theCaldeira-Legget model; for a review see Ref. [132]. Inthe Born-Markov approximation the coupling to the heatbath results in a damping of the density matrix, whichis assumed to be different in the tunnelling region and inthe region of free motion and characterized by the damp-ing coefficients γJ and γC , respectively.

The results for both the average current and the noiseare strongly dependent on the products γJ t0 and γCtC ,where tC is the time of free motion between the leads.Strong decoherence occurs if γJ t0 À 1 or γCtC À 1, thedetails being dependent on the ratio γJ t0/γCtC . Natu-rally, with strong decoherence the phase dependent con-tribution to S(0) is exponentially suppressed since itcomes from correlations over times larger than one os-cillation period. At γJ t0 À 1 the zero-frequency noise isgiven by the expression

S(0) =ω0e

2

π

2γJEJ

E2J + γ2

J

.

This contribution is due to the damped oscillations in thecontact regions (L,R).

In the case of weak damping, γJ t0 ¿ γCtC ¿ 1, onefinds

I =eω0

πtanh

(EJ

kBT

)(cosΦ + cos 2χ) tanh ϑ sinΦ

1 + cosΦ cos 2χ,

S =e2ω0

π

1γCtC

tanh2 ϑ sin2 Φ1 + cosΦ cos 2χ

, (34)

which shows a rich structures as a function of the phasesϑ and χ.

The full counting statistics of Cooper pair transfer wasconsidered by Romito and Nazarov [133]. The authorsfocus on the incoherent regime, where coherence is sup-pressed by classical fluctuations in the gate voltage andno net supercurrent is shuttled. However, charge trans-fers occur, and the current is zero only in average. Thusthe full counting statistics provides a convenient methodto reveal this circumstance.

The basic model for the superconducting shuttle inRef. [133] is similar to that of Refs. [101, 102]. Fluc-tuations of the gate voltage are allowed for by assumingstochastic “white noise” fluctuations,

〈Vg(t)〉 = Vg ,⟨Vg(t)Vg(t′)− V 2

g

⟩=

γ~2

4e2δ(t− t′) .

Thus defined, γ has the meaning of a decoherence rate forthe two charge states. It leads to a damping of the off-diagonal elements of the density matrix, while diagonalelements are assumed to be undamped. The full count-ing statistics is computed using the generating functionmethod.

The physics of charge transfer could be clearly under-stood for the limiting cases of long and short cycles com-paring to the decoherence time. If the shuttling periodis sufficiently long for decoherence to be accomplished,

t−1C , t−1

0 ¿ γ ¿ EJ/~ ,

the full counting (FC) statistics can be interpreted interms of classical elementary events: Cooper pair trans-fers. During the shuttling cycle, either no transfertakes place or one pair is transferred in either direc-tion. There is an apparent similarity with the FCstatistics of the pumping in normal systems studied inRefs. [116, 134, 135]. In this case

p0 = 1/2 , p±2 = 1/4 ,

so that, each shuttling between the superconductorstransfers either one Cooper pair or none, this occurs withequal probability. The pair is transferred with equalprobabilities in either direction. This simple result isquite general and relies neither on the periodicity of shut-tling nor the concrete time dependence of EJ(t) providedadiabaticity is preserved. Leading corrections to the adi-abatic FC statistics are exponentially small, ∼ e−2γt0 .

Adiabaticity is also preserved for small Josephson cou-plings where EJ ¿ ~γ provided

t−1C , t−1

0 ¿ γ .

In this case the factor f ≡ e−t0E2J/~2γ can be arbitrary,

and the FC statistics becomes more complicated [133].For finite f all pn 6= 0, but remain positive definite. Inthe adiabatic limit the FC statistics does not depend onthe superconducting phase Φ or the dynamical phases ϑand χ.

Beyond the adiabatic limit, the FC statistics does de-pend on Φ, and a classical interpretation in this case canfail since pn can be negative of even complex. A rel-atively simple treatment is possible in the case of veryshort shuttling periods, γ/ω0 ¿ 2π. The FC statisticsin this case corresponds to a supercurrent that randomlyswitches between the values ±Is on the time scale 1/γ.The quantities γ and Is depend on the phases Φ, ϑ, andχ, the detailed form of these dependencies being given inRef. [133], see also Ref. [136].

To summarize, in the limiting cases of long and shortshuttling periods the full counting statistics allows forrelative simple classical interpretations. In an intermedi-ate situation, the FC statistics can not be interpreted inclassical terms since the charge transfer probabilities percycle may be negative or complex. This is a clear signa-ture of the fact that superconducting coherence survivesstrong dephasing although this coherence does not man-ifest itself in a net superconducting current.

One can conclude that both the noise spectrum (sec-ond cumulant) and the full counting statistics providevaluable information about shuttle transport, which iscomplimentary to the information that can be extracted

34

from the average current. It is a combination of the fea-tures of the average current and the noise that can assurethat shuttling can be identified as the underlying trans-port mechanism.

6. DISCUSSION AND CONCLUSION

While designing nanometer-sized devices one in-evitably has to face the effects of Coulomb correlationson the electron transport properties. The most peculiarfeature of such correlations is known as single-electrontunnelling, which determines the transport properties ofmany interesting nanodevices. Furthermore, in nanosys-tems electric charge produce not only large potentialdifferences, but also large mechanical forces which canbe comparable with interatomic forces in solids. Theseforces tend to produce mechanical displacements which,in turn, lead to a feedback to the distribution of elec-tric charges. As a result, coupling of electrical and me-chanical degrees of freedom is a hallmark of nanodevices.The aim of this review is to demonstrate one fundamen-tal manifestation of such coupled motion – the shuttletransfer of charge due to the conveying of electrons bya movable part of the nanosystem. Shuttling of chargecan occur either due to an intrinsic instability, or canbe driven by an external AC source. From an “applied”point of view, the role of shuttling can be either positive,or negative. Indeed, it can hinder a proper operation of asingle-electron transistor at the nanometer scale. On theother hand, an intentionally periodic mechanical motionresulting from a designed instability can be used to cre-ate building blocks for new applications. In particular,new principal possibilities for generators and sensors atnanometer scale appear.

As we have tried to show, the research area centeredaround the shuttle instability involves several new princi-ples and possibilities. Thus, there is a wealth of interest-ing physics to be explored containing both coherent andincoherent electron transport facilitated by either clas-sical or quantum mechanical motion. In particular, onecan expect very interesting physics regarding the coher-ent shuttling of Cooper pairs over relatively large dis-tances, as well as the creation of quantum coherence be-tween remote objects by movable superconducting grains.This system, if realized experimentally, would allow for adetermination of the decoherence rate of superconducting

devices due to their interaction with environment.One can imagine several concrete systems where elec-

tromechanical coupling is very important. Among themare nanoclusters or single molecules which can vibrate be-tween leads they bind to, metal-organic composites show-ing pronounced heteroelastic properties, colloidal parti-cles, etc. The likelihood that a similar physical pictureis relevant for the coupling of magnetic and mechanicaldegrees of freedom will certainly lead to new phenomenaand devices. In the latter case the coupling is due toexchange forces, and it can lead to shuttling of magneti-zation.

There are a few experiments where electromechanicalcoupling has been observed and some evidence in favorof single-electron shuttling was presented. The completeexperimental proof of the single-electron shuttle insta-bility remains still a challenging problem. To solve thisproblem in a convincing way it seems to be a good idea tostudy the anomalous structure of the Coulomb blockadein nanomechanical structures with and without gates, aswell as to detect periodic AC currents.

Both experimental and theoretical studies of shuttlecharge transfer are under development, and new worksregularly emerge. In particular, a general approach toshuttling based on an analysis of Breit-Wigner reso-nances in an electronic circuit was developed in a recentpreprint [137].

To summarize, movable nanoclusters can serve as newweak links between various normal, superconducting andmagnetic systems leading to new functionalities of nanos-tructures.

Acknowledgments

This work was supported in part by the EuropeanCommission through project FP6-003673 CANEL of theIST Priority. The views expressed in this publication arethose of the authors and do not necessarily reflect theofficial European Commission’s view on the subject. Fi-nancial support from the Swedish Foundation for Strate-gic Research, from the Swedish Research Council andfrom the U.S. Department of Energy Office of Sciencethrough contract No. W-31-109-ENG-38 is also grate-fully acknowledged. Discussions with V. I. Kozub havebeen greatly appreciated.

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