Electronic spin working mechanically

R. I. Shekhter,1 L. Y. Gorelik,2 I. V. Krive,3, 4 M. N. Kiselev,5

S. I. Kulinich,3 A. V. Parafilo,3 K. Kikoin,6 and M. Jonson1, 7, 8

1Department of Physics, University of Gothenburg, SE-412 96 Goteborg, Sweden2Department of Applied Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden

3B. Verkin Institute for Low Temperature Physics and Engineering of the National

Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine4Physical Department, V. N. Karazin National University, Kharkov 61077, Ukraine

5The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 1-34151 Trieste, Italy6School of Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israel

7SUPA, Institute of Photonics and Quantum Sciences,

Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK8Department of Physics, Division of Quantum Phases and Devices, Konkuk University, Seoul 143-701, Korea

(Dated: January 14, 2015)

A single-electron tunneling (SET) device with a nanoscale central island that can move with re-spect to the bulk source- and drain electrodes allows for a nanoelectromechanical (NEM) couplingbetween the electrical current through the device and mechanical vibrations of the island. Althoughan electromechanical “shuttle” instability and the associated phenomenon of single-electron shuttlingwere predicted more than 15 years ago, both theoretical and experimental studies of NEM-SET struc-tures are still carried out. New functionalities based on quantum coherence, Coulomb correlationsand coherent electron-spin dynamics are of particular current interest. In this article we present ashort review of recent activities in this area.

PACS numbers: 73.23.-b, 72.10.Fk, 73.23.Hk, 85.85.+j

I. INTRODUCTION

Electric weak links play a crucial role in modern nano-electronics since they offer a natural way to inject elec-trons into small conducting areas. At the same time weaklinks of nanometer size offer new functionality due to themesoscopic properties of the small conductors that formsuch links. Coulomb blockade of tunneling, resonant tun-neling, quantum spin coherence, spin-dependent tunnel-ing and weak superconductivity are just examples of newphenomena (compared to bulk transport phenomena)that lead to new physics in nanometer sized weak electriclinks. Special interest is focused on the non-equilibriumevolution of “hot” electrons with voltage-controllable ex-cess energy. Point contact spectroscopy of elementaryexcitations and nanoelectromechanical shuttle instabili-ties are the brightest examples of functionalities based onproperties of accelerated electrons in point contacts. Thenon-equilibrium nature of an electronic system is mostprominently manifested if excitation modes, which arespatially localized in the vicinity of a weak link, interactwith the “hot” electrons. Then even a low level of energytransfer from the electrons does not prevent these excita-tions from accumulating a significant amount of energy,with the energized electrons acting as power supply.

Single-electron tunneling (SET) transistors are nan-odevices with particularly prominent mesoscopic fea-tures. Here, the Coulomb blockade of single-electrontunneling at low voltage bias and temperature [1] makesOhm’s law for the electrical conductance invalid in thesense that the electrical current is not necessarily propor-tional to the voltage drop across the device. Instead, the

current is due to a temporally discrete set of events whereelectrons tunnel quantum-mechanically one-by-one froma source to a drain electrode via a nanometer size island(a “quantum dot”). This is why the properties of a singleelectronic quantum state are crucial for the operation ofthe entire device.Since the probability for quantum mechanical tunnel-

ing is exponentially sensitive to the tunneling distance, itfollows that the position of the quantum dot relative tothe electrodes is crucial. On the other hand the strongCoulomb forces that accompany the discrete nanoscalecharge fluctuations, which are a necessary consequenceof a current flow through the SET device, might cause asignificant deformation of the device and move the dot,hence giving rise to a strong electro-mechanical coupling.This unique feature makes the so-called nanoelectrome-chanical SET (NEM-SET) devices, where mechanical de-formation can be achieved along with electronic oper-ations, to be one of the best nanoscale realizations ofelectromechanical transduction.In this review we will discuss some of the latest achieve-

ments in the nano-electromechanics of NEM-SET devicesfocusing on the new functionality that exploits the co-herence of quantum charge and spin subsystems in theirinterplay with mechanical subsystem. By choosing mag-nets as components of the device one may, take advan-tage of a macroscopic ordering of electrons with respectto their spin. We will discuss how the electronic spincontribute to electromechanical and mechano-electricaltransduction in a NEM-SET device. New effects appearalso due to many-body reconstruction of the electronspectrum in the metallic leads related to exchange in-teraction with spin localized in the moving shuttle. This

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interaction opens a new channel of Kondo resonance tun-neling between the shuttle and the leads, which con-tributes to specific ”Kondo- nano-mechanics”.This review is an update of our earlier reviews of shut-

tling [2–4]. Other aspects of nanoelectromechanics areonly briefly discussed here. We refer readers to the well-known reviews of Refs. 5–9 on nanoelectromechanical sys-tems for additional information.

II. SHUTTLING OF SINGLE ELECTRONS

A single-electron shuttle can be considered as the ul-timate miniaturization of a classical electric pendulumcapable of transferring macroscopic amounts of chargebetween two metal plates. In both cases the electricforce acting on a charged “ball” that is free to move ina potential well between two metal electrodes kept atdifferent electrochemical potentials, eV = µL − µR, re-sults in self-oscillations of the ball. Two distinct physicalphenomena, namely the quantum mechanical tunnelingmechanism for charge loading (unloading) of the ball (inthis case more properly referred to as a grain) and theCoulomb blockade of tunneling, distinguish the nanoelec-tromechanical device known as a single-electron shuttle[10] (see also [11]) from its classical textbook analog. Theregime of Coulomb blockade realized at bias voltages andtemperatures eV, T ≪ EC (where EC = e2/2C is thecharging energy, C is the grain’s electrical capacitance)allows one to consider single electron transport throughthe grain. Electron tunneling, being extremely sensitiveto the position of the grain relative to the bulk electrodes,leads to a shuttle instability — the absence of any equi-librium position of an initially neutral grain in the gapbetween the electrodes.

A. Shuttle instability in the quantum regime of

Coulomb blockade

At first, we consider the single-electron shuttle effectin the simplest model [12] where the grain is modeled asa single-level quantum dot (QD) that is weakly coupled(via a tunnel Hamiltonian) to the electrodes (see Fig. 1).The Hamiltonian corresponding to this model reads

Htot =∑

j=L,R

H(j)l +HQD +Hv +

∑

j=L,R

H(j)t , (1)

where the Hamiltonian

H(j)l =

∑

k

(εkj − µj)a†kjakj (2)

describes noninteracting electrons in the left (j = L) andright (j = R) leads, which are kept at different chemi-cal potential µj and have a constant density states νj ;

x

µ eVL =

µR = 0

LeadLead

Dot

FIG. 1: Model system consisting of a movable quantum dotplaced between two leads. An effective elastic force acting onthe dot due to its connections to the leads is described by aparabolic potential. Only one single electron state is availablein the dot and the non-interacting electrons in the leads areassumed to have a constant density of states. Reprinted withpermission from [12], D. Fedorets et al., Europhys. Lett. 58,99 (2002). c© 2002, EDP Sciences.

a†kj(akj) creates (annihilates) an electron with momen-tum k in lead j. The quantum dot is described by twoparts. It is single electron level Hamiltonian and Hamil-tonian of harmonic potential in which QD vibrates

HQD = ε0c†c− dxc†c, (3)

Hv =1

2(x2 + p2), (4)

where c†(c) is the creation (annihilation) operator foran electron at the dot, ε0 is the energy of the resonantlevel, x is the dimensionless coordinate operator (nor-malized by the amplitude x0 of zero-point fluctuations,x0 =

√h/Mω0, M is the mass of QD), p is the cor-

responding momentum operator ([x, p] = i), ω0 is thefrequency of vibrons, d = eE/(Mω2

0x0) is characteristicelectromechanical interaction constant. For conveniencewe use dimensionless variables. The physical meaning ofthe second term in Eq. (3) for usual shuttle systems isthe interaction energy due to the coupling of the electroncharge density on the dot with the electric field (E) inthe gap between electrodes. Here, for convenience, all en-ergies measure in units hω0, time in units of ω−1

0 . Note,that in general the mechanism of electromechanical inter-action could have different nature (electrostatic interac-tion charge on the dot with gate electrode, interaction inmagnetic field due the Lorentz force, due exchange forcebetween electrons with spin and spin polarized leads, seenext Sections).

The tunneling Hamiltonian H(j)t in Eq. (1) has the

form:

H(j)t =

∑

k

t0j exp(jx/λ)a†kjc+ h.c. (5)

3

Here j = ± for L/R electrodes, t0j is the bare tunnelingamplitude, which corresponds to a weak dot-electrodecoupling, λ is the characteristic tunneling length. Theexplicit coordinate dependence in the tunneling Hamilto-nian indicates sensitivity of tunnel matrix elements to ashift of the quantum dot center-of-mass coordinate withrespect to its equilibrium (xcm = 0) position. The x-dependence in Eq. (5) represents also additional interac-tion with vibronic degree of freedom.Even in such a simple formulation the single-electron

shuttle problem is quite complex. In this section we re-view some main results of electron shuttling (without in-volving the spin degree of freedom) and present the basicidea of solution’s method based on the equation of motionfor the matrix density. The advantage of this method isthat it is possible to explicitly consider the quantum dotdynamics in quantum regime and take into account thecoherent dynamics of spin electron’s states in a magneticfield, see the next section.The time evolution of the system is obtained from the

Liouville-von Neumann equation for the total density ma-trix

ih∂tσ(t) = [H, σ(t)]. (6)

In order to consider the dynamics of the electronic statein the dot and the vibronic degrees of freedom we reducethe total density operator by tracing over all electronicstates in the leads, ρ(t) = Trleadsσ(t). We assume thatelectrons in the leads are in equilibrium and that they arenot affected by the coupling to the dot. So, we factorizethe density matrix, σ(t) ≈ ρ(t)⊗ σleads (this approxima-tion is always valid for Γj = 2πνj |t0j |2 exp[∓x/λ] ≪ 1).After shifting the x-axis by d/2 we get the system ofequation of motion for the diagonal elements of densitymatrix ρ0 = 〈0|ρ|0〉 and ρ1 = 〈1|ρ|1〉, where |1〉 = d†|0〉,as

∂tρ0 = −i

[(Hv +

d

2x, ρ0

]− 1

2ΓL(x), ρ0

+

√ΓR(x)ρ1

√ΓR(x), (7)

∂tρ1 = −i

[(Hv −

d

2x, ρ1

]− 1

2ΓR(x), ρ1

+

√ΓL(x)ρ0

√ΓL(x), (8)

where Γj(x) = Γj(x+d/2)]. The off-diagonal density ma-trix elements are decoupled from the equation of motionof the diagonal elements. It is easy to take into accountdissipation of the system. The corresponding dissipationterm is Lγρ = −(iγ/2)[x, p, ρ]−(γ/2)[x, [x, p]] (γ is thedissipation rate).Now we find the condition under which the vibrational

ground state of the oscillator becomes unstable. For thiswe consider the time evolution of the expectation valueof the coordinate, x(t) = Trxρ+, and the momentum

operators, p(t) = Trpρ+, of the island (here ρ+ ≡ρ0 + ρ1). To first order in λ−1, for symmetric tunneling

couplings ΓL(0) = ΓR(0) = Γ/2 and in the high biasvoltage limit (µL − µR = eV → ∞) the equations ofmotion for the first vibrational moments become closed,so that [13]

˙x = p, ˙p = −γp− x− d

2n−, (9)

n− = −Γn− +2Γ

λx,

where n− = 1 − 2Trρ1. The solution of Eq. (9) for thequantum dot displacement is x(t) ≈ Aert cos(t), wherer = 1/2(γthr − γ) is the rate of increment of the shuttleinstability. If the dissipation rate γ is below the thresholdvalue γthr = Γd/[λ(Γ2) + 1], then the expectation valueof the dot coordinate grows exponentially in time and thevibrational ground state is unstable. It was shown [13]that this exponential increase of the displacement drivesthe system into the nonlinear regime of the vibration dy-namics, where the system reaches a stable steady stateof developed shuttle motion.

In order to analyze this stable state (i.e. the solutionof the system Eq. (7,8) it is convenient to use the Wignerfunction representation [14], [13]. The Wigner distribu-tion function for the density operator ρ+ is defined as

W+(x, p) ≡1

2π

∫ +∞

−∞

dξe−ipξ 〈x+ ξ/2|ρ+|x− ξ/2〉 .(10)

The dynamics of the oscillating QD is characterized byits trajectory (distribution) in the phase space (x, p) forp2/2 + x2/2 = const. Now we proceed to polar coordi-nates (A,ϕ), where x = A sinϕ and p = A cosϕ. Anequation for W+(A,ϕ) is derived from Eqs. (7) and (8)after straightforward calculations (for details see [13]).To leading order in the small parameters d/λ, λ−2, andγ this equation takes the form of a stationary Fokker-Planck equation for the zeroth Fourier component of theWigner function W+(A)

∂

∂A

(D0(A)

∂

∂A−D1(A)

)W+(A) = 0, (11)

where D1 = A2D1(A), D0 = AD0(A) are drift- and diffu-sion coefficients (analytical expression of this coefficientswill be presented in section for the magnetic shuttle).The normalized solution of Eq. (11) has the form of aBoltzman distribution,

W+ = Z−1 exp

(∫ A

0

dAD1(A)

D0(A)

)(12)

The stationary solution of the oscillating dot is localizedin the phase space around points where W+ is maximal.From Eq. (12) one can see that the maximum of theWigner function is determined by zeros of the drift coef-

4

ficient D1(Am) = 0 (D′

1(Am) < 0). In the vicinity of thispoint, W+ can be approximated by a Gaussian distribu-tion function. For the spinless shuttle problem it can beshown that W+ always has an extremum at A = 0: max-imum for γ > γthr and minimum for γ < γthr. So thevibrational ground state is unstable when the dissipationis below threshold value as has been shown by solving theequation system (9). The function W+ has also a maxi-mum for the non-zero amplitude AC , which correspondsto the stable limit cycle amplitude of shuttle oscillations(for more details see [13]).One can distinguish two regimes of ”quantum” (for

d/λ ≪ λ−4) and ”quasiclassical” (d/λ ≫ λ−4) shuttlemotion. In the quasiclassical regime Gaussian distribu-tion is narrow and in quantum regime the width of dis-tribution “bell” is of the order of λ ≫ 1, i.e. the Wignerfunction is smeared around classical phase trajectory. Itis interesting to note that a region of parameters existswhere both vibrational and shuttling regimes are present(a region where the Wigner function has two maxima).

III. ELECTRO - AND SPINTRO - MECHANICS

OF MAGNETIC SHUTTLE DEVICES

In this Section we will explore new functionalities thatemerge when nanomechanical devices are partly or com-pletely made of magnetic materials. The possibility ofmagnetic ordering brings new degrees of freedom intoplay in addition to the electronic and mechanical onesconsidered so far, opening up an exciting perspective to-wards utilising magneto-electro-mechanical transductionfor a large variety of applications. Device dimensions inthe nanometer range mean that a number of mesoscopicphenomena in the electronic, magnetic and mechanicalsubsystems can be used for quantum coherent manipu-lations. In comparison with the electromechanics of thenanodevices considered above the prominent role of theelectronic spin in addition to the electric charge shouldbe taken into account.The ability to manipulate and control spins via electri-

cal [16–18] magnetic [19] and optical [20, 21] means hasgenerated numerous applications in metrology [22] in re-cent years. A promising alternative method for spin ma-nipulation employs a mechanical resonator coupled to themagnetic dipole moment of the spin(s), a method whichcould enable scalable quantum information architectures[23] and sensitive nanoscale magnetometry [24–26]. Mag-netic resonance force microscopy (MRFM) was suggestedas a means to improve spin detection to the level ofa single spin and thus enable three dimensional imag-ing of macromolecules with atomic resolution. In thistechnique a single spin, driven by a resonant microwavemagnetic field interacts with a ferromagnetic particle.If the ferromagnetic particle is attached to a cantilevertip, the spin changes the cantilever vibration parameters[27]. The possibility to detect [27] and monitor the co-herent dynamics of a single spin mechanically [28] has

been demonstrated experimentally. Several theoreticalsuggestions concerning the possibility to test single-spindynamics through an electronic transport measurementwere made recently [29–32]. Complementary studies ofthe mechanics of a resonator coupled to spin degrees offreedom by detecting the spin dynamics and relaxationwere suggested in [29–36] and carried out in [37]. Elec-tronic spin-orbit interaction in suspended nanowires wasshown to be an efficient tool for detection and coolingof bending-mode nanovibrations as well as for manipu-lation of spin qubit and mechanical quantum vibrations[38–40].An obvious modification of the nano-electro-mechanics

of magnetic shuttle devices originates from the spin-splitting of electronic energy levels, which results in theknown phenomenon of spin-dependent tunneling. Spin-controlled nano-electro-mechanics which originates fromspin-controlled transport of electric charge in magneticNEM systems is represented by number of new magneto-electro-mechanical phenomena.Qualitatively new opportunities appear when magnetic

nanomechanical devices are used. They have to do withthe effect of the short-ranged magnetic exchange inter-action between the spin of electrons and magnetic partsof the device. In this case the spin of the electron ratherthan its electrical charge can be the main source of themechanical force acting on movable parts of the device.This leads to new physics compared with the usual elec-tromechanics of non-magnetic devices, for which we usethe term spintro-mechanics. In particular it becomes pos-sible for a movable central island to shuttle magnetiza-tion between two magnetic leads even without any chargetransport between the leads. The result of such a me-chanical transportation of magnetization is a magneticcoupling between nanomagnets with a strength and signthat are mechanically tunable.In this Section we will review some early results that

involve the phenomena mentioned above. These onlyamount to a first step in the exploration of new oppor-tunities caused by the interrelation between charge, spinand mechanics on a nanometer length scale.

A. Spin-controlled shuttling of electric charge

By manipulating the interaction between the spin ofelectrons and external magnetic fields and/or the inter-nal interaction in magnetic materials, spin-controlled na-noelectromechanics may be achieved.A new functional principle — spin-dependent shuttling

of electrons — for low magnetic field sensing purposeswas proposed by Gorelik et al. in Ref. 41. This principlemay lead to a giant magnetoresistance effect in externalmagnetic fields as low as 1-10 Oe in a magnetic shuttle de-vice if magnets with highly spin-polarized electrons (halfmetals [42–46]) are used as leads in a magnetic shuttle de-vice. The key idea is to use the external magnetic field tomanipulate the spin of shuttled electrons rather than the

5

magnetization of the leads. Since the electron spends arelatively long time on the shuttle, where it is decoupledfrom the magnetic environment, even a weak magneticcan rotate its spin by a significant angle. Such a rota-tion allows the spin of an electron that has been loadedonto the shuttle from a spin-polarized source electrodeto be reoriented in order to allow the electron finally totunnel from the shuttle to the (differently) spin-polarizeddrain lead. In this way the shuttle serves as a very sen-sitive “magnetoresistor” device. The model employed inRef. 41 assumes that the source and drain are fully po-larized in opposite directions. A mechanically movablequantum dot (described by a time-dependent displace-ment x(t)), where a single energy level is available forelectrons, performs driven harmonic oscillations betweenthe leads. The external magnetic field, H, is perpendicu-lar to the orientations of the magnetization in both leadsand to the direction of the mechanical motion.The spin-dependent part of the Hamiltonian is speci-

fied as

Hmagn(t) = J(t)(a†↑a↑−a†↓a↓)−gµH

2(a†↑a↓+a†↓a↑), (13)

where J(t) = JR(t) − JL(t), JL(R)(t) are the molecu-lar fields induced by exchange interactions between theon-grain electron and the left(right) lead, g is the gyro-magnetic 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.In the limit of weak exchange field, Jmax ≪ µH one

may neglect the influence of the magnetic leads on theon-dot electron spin dynamics. The resulting current is

I =eω0

π

sin2(ϑ/2) tanh(w/4)

sin2(ϑ/2) + tanh2(w/4)(14)

where w is the total tunneling probability during the con-tact time t0, while ϑ ∼ πgµH/hω0 is the rotation angleof the spin during the “free-motion” time.The theory [41] predicts oscillations in the magnetore-

sistance of the magnetic shuttle device with a period∆Hp, which is determined from the equation hω0 =gµ(1 + w)∆Hp. The physical meaning of this relationis simple: every time when ω0/Ω = n+1/2 (Ω = gµH/his the spin precession frequency in a magnetic field) theshuttled electron is able to flip fully its spin to removethe “spin-blockade” of tunneling between spin polarizedleads having their magnetization in opposite directions.This effect can be used for measuring the mechanical fre-quency thus providing dc spectroscopy of nanomechani-cal vibrations.Spin-dependent shuttling of electrons as discussed

above is a property of non-interacting electrons, in thesense that tunneling of different electrons into (and outof) the dot are independent events. The Coulomb block-ade phenomenon adds a strong correlation of tunnelingevents, preventing fluctuations in the occupation of elec-

tronic states on the dot. This effect crucially changesthe physics of spin-dependent tunneling in a magneticNEM device. One of the remarkable consequences isthe Coulomb promotion of spin-dependent tunneling pre-dicted in Ref. 47. In this work a strong voltage depen-dence of the spin-flip relaxation rate on a quantum dotwas demonstrated. Such relaxation, being very sensi-tive to the occupation of spin-up and spin-down stateson the dot, can be controlled by the Coulomb block-ade phenomenon. It was shown in Ref. 47 that by lift-ing the Coulomb blockade one stimulates occupation ofboth spin-up and spin-down states thus suppressing spin-flip relaxation on the dot. In magnetic devices withhighly spin-polarized electrons electronic spin-flip can bethe only mechanism providing charge transport betweenoppositely magnetized leads. In this case the onset ofCoulomb blockade, by increasing the spin-flip relaxationrate, stimulates charge transport through a magneticSET device (Coulomb promotion of spin-dependent tun-neling). Spin-flip relaxation also modifies qualitativelythe noise characteristics of spin-dependent single-electrontransport. In Refs. 48, 49 it was shown that the low-frequency shot noise in such structures diverges as thespin relaxation rate goes to zero. This effect provides anefficient tool for spectroscopy of extremely slow spin-fliprelaxation in quantum dots. Mechanical transportationof a spin-polarized dot in a magnetic shuttle device pro-vides new opportunities for studying spin-flip relaxationin quantum dots. The reason can be traced to a spin-blockade of the mechanically aided shuttle current thatoccurs in devices with highly polarized and colinearlymagnetized leads. As was shown in Ref. 50 the aboveeffect results in giant peaks in the shot-noise spectralfunction, wherein the peak heights are only limited bythe rates of electronic spin flips. This enables a nanome-chanical spectroscopy of rare spin-flip events, allowingspin-flip relaxation times as long as 10 µs to be detected.

The spin-dependence of electronic tunneling in mag-netic NEM devices permits an external magnetic fieldto be used for manipulating not only electric transportbut also the mechanical performance of the device. Thiswas demonstrated in Refs. 51, 52. A theory of the quan-tum coherent dynamics of mechanical vibrations, electroncharge and spin was formulated and the possibility totrigger a shuttle instability by a relatively weak magneticfield was demonstrated. It was shown that the strengthof the magnetic field required to control nanomechanicalvibrations decreases with an increasing tunnel resistanceof the device and can be as low as 10 Oe for giga-ohmtunnel structures.

A new type of nanoelectromechanical self excitationcaused entirely by the spin splitting of electronic en-ergy levels in an external magnetic field was predictedin Ref. 54 for a suspended nanowire, where mechanicalmotion in a magnetic field induces an electromotive cou-pling between electronic and vibrational degrees of free-dom. It was shown that a strong correlation betweenthe occupancy of the spin-split electronic energy levels in

6

(a)

M

(b)

FIG. 2: A movable quantum dot in a magnetic shuttle de-vice can be displaced in response to two types of force: (a) along-range electrostatic force causing an electromechanical re-sponse if the dot has a net charge, and (b) a short-range mag-netic exchange force leading to “spintromechanical” responseif the dot has a net magnetization (spin). The direction ofthe force and displacements depends on the relative signs ofthe charge and magnetization, respectively. Reprinted withpermission from [59], R. I. Shekhter et al., Phys. Rev. B 86,100404 (2012). c© 2012, American Physical Society.

the nanowire and the velocity of flexural nanowire vibra-tions provides energy supply from the source of DC cur-rent, flowing through the wire, to the mechanical vibra-tions thus making possible stable, self-supporting bend-ing vibrations. Estimations made in Ref. 54 show thatin a realistic case the vibration amplitude of a suspendedcarbon nanotube (CNT) of the order of 10 nm can beachieved if magnetic field of 10 T is applied.

B. Spintro-mechanics of magnetic shuttle devices

New phenomena, qualitatively different from the elec-tromechanics of nonmagnetic shuttle systems, may ap-pear in magnetic shuttle devices in a situation whenshort-range magnetic exchange forces become compara-ble in strength to the long-range electrostatic forces be-tween the charged elements of the device [54]. There isconvincing evidence that the exchange field can be severaltesla at a distance of a few nanometers from the surfaceof a ferromagnet [55–58]. Because of the exponential de-cay of the field this means that the force experienced by asingle-electron spin in the vicinity of magnetic electrodescan be very large. These spin-dependent exchange forcescan lead to various “spintro-mechanical” phenomena.Mechanical effects produced by a long-range electro-

static force and short-ranged exchange forces on a mov-able quantum dot are illustrated in Fig. 2. The elec-trostatic force acting on the dot, placed in the vicinityof a charged electrode (Fig. 2(a)), is determined by theelectric charge accumulated on the dot. In contrast, theexchange force induced by a neighboring magnet dependson the net spin accumulated on the dot. While the elec-trostatic force changes its direction if the electric chargeon the dot changes its sign, the spin-dependent exchangeforce is insensitive to the electric charge but it changesdirection if the electronic spin projection changes its sign.

A very important difference between the two forces is thatthe electrostatic force changes only as a result of injec-tion of additional electrons into (out of) the dot while thespintronic force can be changed due to the electron spindynamics even for a fixed number of electrons on the dot(as is the case if the dot and the leads are insulators). Inthis case interesting opportunities arise from the possibil-ity of transducing the dynamical variations of electronicspin (induced, e.g., by magnetic or microwave fields) tomechanical displacements in the NEM device. In Ref. 59a particular spintromechanical effect was discussed – a gi-ant spin-filtering of the electron current (flowing throughthe device) induced by the formation of what we shallcall a “spin-polaronic state”.

The Hamiltonian that describes the magnetic nanome-chanical SET device in Ref. 59 has the standard form (itsspin-dependent part depends now on the mechanical dis-placement of the dot). HenceH = Hlead+Htunnel+Hdot,

where Hleads =∑

k,σ,s εksσa†ksσaksσ describes electrons

(labeled by wave vector k and spin σ =↑, ↓) in the twoleads (s = L,R). Electron tunneling between the leadsand the dot is modeled as

Htunnel =∑

k,σ,s

Ts(x)a†ksσcσ +H.c. (15)

where the matrix elements Ts(x) = T(0)s exp(∓x/λ) (λ

is the characteristic tunneling length) depend on the dotposition x. The Hamiltonian of the movable single-leveldot is

Hdot = hω0b†b+

∑

σ

nσ[ε0−sgn(σ)J(x)]+UCn↑n↓, (16)

where sgn(↑, ↓) = ±1, UC is the Coulomb energy associ-ated with double occupancy of the dot and the eigenval-ues of the electron number operators nσ is 0 or 1. Theposition dependent magnitude J(x) of the spin dependentshift of the electronic energy level on the dot is due to theexchange interaction with the magnetic leads. Here weexpand J(x) to linear order in x so that J(x) = J (0)+ jxand without loss of generality assume that J (0) = 0.

The modification of the exchange force, caused bychanging the spin accumulated on the dot, shifts the equi-librium position of the dot with respect to the magneticleads of the device. Since the electron tunneling ma-trix element is exponentially sensitive to the position ofthe dot with respect to the source and drain electrodesone expects a strong spin-dependent renormalization ofthe tunneling probability, which exponentially discrim-inates between the contributions to the total electricalcurrent from electrons with different spins. This spatialseparation of dots with opposite spins is illustrated inFig. 3. While changing the population of spin-up andspin-down levels on the dot (by changing e.g. the biasvoltage applied to the device) one shifts the spatial posi-tion x of the dot with respect to the source/drain leads.It is important that the Coulomb blockade phenomenon

7

0

1

2

Spatial position of the dot

Gra

in p

opula

tion

2D

FIG. 3: Diagram showing how the equilibrium position ofthe movable dot depends on its net charge and spin. Thedifference in spatial displacements discriminates transportthrough a singly occupied dot with respect to the electronspin. Reprinted with permission from [59], R. I. Shekhter et

al., Phys. Rev. B 86, 100404 (2012). c© 2012, AmericanPhysical Society.

prevents simultaneous population of both spin states. Ifthe Coulomb blockade is lifted the two spin states becomeequally populated with a zero net spin on the dot, S = 0.This removes the spin-polaronic deformation and the dotis situated at the same place as a non-populated one.In calculations a strong modification of the vibrationalstates of the dot, which has to do with a shift of its equi-librium position, should be taken into account. This re-sults in a so-called Franck-Condon blockade of electronictunneling [60, 61]. The spintro-mechanical stimulation ofa spin-polarized current and the spin-polaronic Franck-Condon blockade of electronic tunneling are in compe-tition and their interplay determines a non-monotonicvoltage dependence of the giant spin-filtering effect.

To understand the above effects in more detail con-sider the analytical results of Ref. 59. A solution of theproblem can be obtained by the standard sequential tun-neling approximation and by solving a Liouville equationfor the density matrix for both the electronic and vibronicsubsystems. The spin-up and spin-down currents can beexpressed in terms of transition rates (energy broaden-ing of the level) and the occupation probabilities for thedot electronic states. For simplicity we consider the caseof a strongly asymmetric tunneling device. At low biasvoltage and low temperature the partial spin current is

Iσ ∼ eΓL

hexp

(1

2

[x20

λ2−(

x0

hω0

)2]− sgn(σ)β

), (17)

where β = x20/hω0λ. In the high bias voltage (or tem-

perature) regime, maxeV, T ≫ Ep, where the pola-ronic blockade is lifted (but double occupancy of the dotis still prevented by the Coulomb blockade), the current

FIG. 4: Spin polarization of the current through the modelNEM-SET device under discussion. Reprinted with permis-sion from [59], R. I. Shekhter et al., Phys. Rev. B 86, 100404(2012). c© 2012, American Physical Society.

expression takes the form

Iσ ∼ eΓL

hexp

([2nB + 1]

x20

λ2− 2 sgn(σ)β

), (18)

where nB is Bose-Einstein distribution function. Thescale of the polaronic spin-filtering of the device is de-termined by the ratio β of the polaronic shift of theequilibrium spatial position of a spin-polarized dot andthe electronic tunneling length. For typical values of theexchange interaction and mechanical properties of sus-pended carbon nanotubes this parameter is about 1-10.As was shown this is enough for the spin filtering of theelectrical current through the device to be nearly 100 %efficient. The temperature and voltage dependence of thespin-filtering effect is presented in Fig. 4. The spin filter-ing effect and the Franck-Condon blockade both occur atlow voltages and temperatures (on the scale of the pola-ronic energy; see Fig. 4 (a)). An increase of the voltageapplied to the device lifts the Franck-Condon blockade,which results in an exponential increase of both the cur-rent and the spin-filtering efficiency of the device. Thisincrease is blocked abruptly at voltages for which theCoulomb blockade is lifted. At this point a double occu-pation of the dot results in spin cancellation and removalof the spin-polaronic segregation. This leads to an ex-ponential drop of both the total current and the spinpolarization of the tunnel current (Fig. 4 (b)). As onecan see in Fig. 4 prominent spin filtering can be achievedfor realistic device parameters. The temperature of oper-ation of the spin-filtering device is restricted from aboveby the Coulomb blockade energy. One may, however,consider using functionalized nanotubes [62] or grapheneribbons [63] with one or more nanometer-sized metal orsemiconductor nanocrystal attached. This may providea Coulomb blockade energy up to a few hundred kelvin,making spin filtering a high temperature effect [59].

C. Spintronics of shuttles

In this subsection we discuss the possibility to ma-nipulate the spin of tunneling electrons by an external

8

magnetic field and how it can affect electron transportthrough a nanoelectromechanical device. In the simplestmodel, we assume that the left and right electrodes arefully spin polarized. The movable single level quantumdot (in the absence of a magnetic field) can vibrate inthe gap between two leads. A bias voltage is applied butelectron transport through the system is blocked sincethe source and drain leads are fully spin polarized inopposite direction. An external magnetic field appliedpendicular to the direction of the magnetization in theelectrode leads to precession of the electron spin of thequantum dot and as a consequence the electron trans-port is unblocked. The Hamiltonian of the system has

the form [52] of Eq. (1) with Hleads = Σjkεj(k)c†jkcjk

(j = L,R → j = (↑, ↓)) and

HQD = (ε0 − dx)∑

σ

c†σcσ − h

2(c†↑c↓ + c†↓c↑) + Uc†↑c↑c

†↓c↓,

(19)where h = gµBH/hω0 is the dimensionless magneticfield. To analyze this system we use the method describedin Section 2. A quantum master equation for the reduceddensity matrix operator ρ0 ≡ 〈0 | ρ | 0〉, ρ↑ ≡ 〈↑| ρ |↑〉,ρ↓ ≡ 〈↓| ρ |↓〉, and ρ↑↓ ≡ 〈↑| ρ |↓〉 is obtained in analogywith the spinless case

∂ρ0∂t

= −i [Hv + xd, ρ0]

−ΓL(x), ρ0

/2 +

√ΓR(x)ρ↓

√ΓR(x), (20)

∂ρ↓∂t

= −i [Hv, ρ↓]

+ih

2(ρ↑↓ − ρ↑↓)−

1

2

Γ+(x), ρ↓

, (21)

∂ρ↑∂t

= −i [Hv, ρ↑]− ih

2(ρ↑↓ − ρ↑↓)

+

√ΓL(x)ρ0

√ΓL(x) +

√ΓR(x)ρ2

√ΓR(x),(22)

∂ρ↑↓∂t

= −i [Hv, ρ↑↓] + ih

2[ρ↓ − ρ↑]−

1

2ρ↑↓Γ+(x),(23)

∂ρ↓↑∂t

= −i [Hv, ρ↓↑]− ih

2[ρ↓ − ρ↑]−

1

2Γ+(x)ρ↑↓,(24)

∂ρ2∂t

= −i [Hv − xd, ρ2]

−ΓR(x), ρ2

/2 +

√ΓL(x)ρ↑

√ΓL(x), (25)

where Γ+(x) = ΓL(x) + ΓR(x). The set of equa-tions (20)-(25) is derived in the high bias voltage limit:eV/2 − ε0 − U ≫ hω0. In general, the problem can besolved in two limits with and without the Coulomb block-ade regime. In the Coulomb blockade regime the secondelectron can not tunnel onto the quantum dot due toCoulomb repulsion. Hence the probability for double oc-cupancy ρ2 → 0. First, we focus on the case withoutCoulomb blockade.

Here we repeat the analysis scheme for the evolution of

the stationary solution W+(A) for the probability of theshuttle to vibrate with an amplitude A. Expanding thefunction D1(A) around A = 0 one can get the conditionfor the shuttle instability γ < γthr = Γ(2h2d)/λ(h2+Γ2).As in the case of spinless electron, the function W+ has amaximum at A = 0 (stable point) when dissipation rateγ is above the threshold value. In the opposite case thevibrational ground state is unstable.The positive bounded function β0(A, h) = (2D1(A) −

γ)λ/d has only one maximum and monotonically de-

creases for large A. In [52] it was shown that if h <√3Γ,

the function β0 has a maximum at A = 0, while forh >

√3Γ, this function has a minimum at A = 0. The

structure of the function β0 determines the behavior ofthe system in the parameter space d−h (or γ−h). Thereare several areas or phases. In the first phase (vibronic),defined by d/γλ < 1/h[maxβ0(A)], the system is in thelowest vibrational state (A = 0 is a stable point). Theshuttle phase is developed when γ < γthr and there isonly one stable point at A 6= 0. The third phase is themixed phase. It appears because the two above phasesbecome unstable if h exceeds the critical value

√3Γ.

In the Coulomb blockade regime the same analysisgives that D1(A) is positive for all values of h if Γ < 4/3.On the other hand, if Γ > 4/3, there is a range ofmagnetic field strenghts where a shuttle instability doesnot occur. In particular, when Γ ≫ 1 this interval is0 < h < Γ/

√2. This implies that in the adiabatic regime

of charge transport (Γ ≫ 1) in weak magnetic field thereis no instability and the electrically driven electron shut-tle is realized only in strong magnetic fields.

D. Electron Shuttle Based on Electron Spin

In the previous subsection we studied the shuttle in-stability in the case of an electromechanical coupling be-tween the quantum dot and the leads. In the Coulombblockade regime a shuttle instability appears if an exter-nal magnetic field h exceeds the critical value hcr =

√3Γ.

Here we will study the shuttle instability in the case whenthe interaction between the dot and the leads is due to amagnetic (exchange) coupling [53].The Hamiltonian of the system is similar to the one

considered in Section III. C. The only difference is thatthe quantum dot Hamiltonian reads

Hdot = ε0(a†↑a↑ + a†↓a↓)

−JL(x)(a†↑a↑ − a†↓a↓)− JR(x)(a

†↓a↓ − a†↑a↑)

−gµH

2(a†↑a↓ + a†↓a↑)− Ua†↑a

†↓a↑a↓ . (26)

In what follows we will consider the symmetrical case,JR(x) = JL(−x) and restrict ourselves to the Coulombblockade regime, U ∼ e2/2C >| eV/2− ε0 |.Following Ref.[52] one gets equations of motion for the

reduced density matrix operators ρ0 ≡ 〈0 | ρ | 0〉, ρ↑ ≡

9

〈↑| ρ |↑〉, ρ↓ ≡ 〈↓| ρ |↓〉, and ρ↑↓ ≡ 〈↑| ρ |↓〉:

∂ρ0∂t

= −i [Hv, ρ0]

−ΓL(x), ρ0 /2 +√ΓR(x)ρ↓

√ΓR(x), (27)

∂ρ↑∂t

= −i [Hv, ρ↑] + i [J(x), ρ↑]

−ih(ρ↑↓ − ρ†↑↓

)/2 +

√ΓL(x)ρ0

√ΓL(x),(28)

∂ρ↓∂t

= −i [Hv, ρ↓]− i [J(x), ρ↓]

+ih(ρ↑↓ − ρ†↑↓

)/2− ΓR(x), ρ↓ /2, (29)

∂ρ↑↓∂t

= −i [Hv, ρ↑↓] + i J(x), ρ↑↓+ih (ρ↓ − ρ↑) /2− ρ↑↓ΓR(x)/2 (30)

In Eqs. (27)-(30) Γj(x) = Γexp(j2x/λ) and J(x) =JL(x) − JR(x). In what follows we assume a linear x-dependence of J(x): J(x) ≃ −αx+ ..., α = 2J ′

R(0) > 0.

The difference between our operator equations and thecorresponding equations in Ref. [52] (rewritten for theCoulomb blockade case) is the appearance of terms in-duced by the coordinate-dependent exchange interactionJ(x). These appear in Eqs. (27)-(30) as a commutatorterm for ρ↑ and ρ↓ and as an anti-commutator term forρ↑↓. In contrast to the electrically driven shuttle, thedriving force in our case is strongly connected to the spindynamics, which results in a completely different depen-dence of the shuttle behavior on magnetic field.

Both linear and nonlinear regimes of the shuttling dy-namics can be conveniently analyzed by using the Wignerfunction representation of the density operators [14].This approach allows one to calculate the Wigner dis-tribution function Wρ(x, p) for the vibrational degree offreedom to lowest order in the small parameters α and1/λ for small (compared to λ) shuttle vibration ampli-

tudes A. The relevant Wigner function, W(0)Σ (A), aver-

aged over the shuttle phase ϕ (x = A sinϕ), solves thestationary Fokker-Planck equation as in Eq. (11) withdrift- and diffusion coefficients containing the factors

D1 =α

λ

h2Γ3

Γ2 + 3h2

3Γ2 + 3− h2

Q0(Γ, h)(31)

D0 =h2Γ

Γ2 + 3h2

[α2Q1(Γ, h) + λ−2Q0(Γ, h)

2Q0(Γ, h)

](32)

respectively, where

Q0(Γ, h) =(1− h2 − 2Γ2

)2+

Γ2

4

(Γ2 + 3h2 − 5

)2, (33)

Q1(Γ, h) =

(1 +

9Γ2

4

)(1 + h2 + 2Γ2

)− 5Γ4

4. (34)

In Eqs. (31)-(34) all energies are normalized with respectto the energy quantum hω of the mechanical vibrations:hω → 1, gµH/hω → h, J(x)/hω → J(x), Γj(x)/ω →

Γj(x) [hΓj(x) = 2πν | Tj(x) |2 are partial level widths].

For A ≪ 1 the solution of Eq. (11) takes the form of

a Boltzmann distribution function, W(0)Σ ∼ exp(−βE),

where E = A2/2 is the dot’s vibrational energy and 1/β,where

β =

(2αΓ2

λ

)h2 − 3Γ2 − 3

α2Q1(Γ, h) + λ−2Q0(Γ, h), (35)

is an effective temperature. Since the functions Q0 andQ1 are positive, the sign of the effective temperature isdetermined by the relation between magnetic field, levelwidth and vibration quantum. In particular the effectivetemperature is negative at small magnetic fields, |H| <Hc, where (reverting to dimensional variables) gµHc =

h√3 (Γ2 + ω2).

A negative β implies that the static state of the dot(A = 0) is unstable and that a shuttling regime of chargetransport (A 6= 0) is realized. It is interesting to notethat β is finite even as h → 0. This apparent paradoxmay be resolved by considering the Fokker-Plank equa-tion in its time-dependent form and noting that the rateof change of the oscillation amplitude at the instabilityis defined by the coefficient D1. This coefficient scalesas D1(h) ∝ h2 as h → 0 and therefore the shuttle phaseis only realized formally after an infinitely long time inthis limit. As a function of magnetic field D1 has a max-imum, Dmax

1 = 0.6(α/λ)Γ−1, at hopt = 0.4Γ. There-fore, optimal magnetic fields are in the range 0.1− 1 T ifhΓ = 10− 100 µeV. For high magnetic fields, |H| > Hc,there is no shuttling regime (at least not with a smallvibration amplitude, A ≪ 1) and the vibronic regime,corresponding to small fluctuations of the quantum dotaround its equilibrium position, is stable.

The amplitude of the shuttle vibrations that developas the result of an instability is still described by Eq. (11)for the Wigner distribution function. However, for largeamplitudes, A >∼ 1, the drift- and diffusion coefficientsA2D1 and AD0 can no longer be evaluated analytically.Fortunately, it is sufficient to know the amplitude- andmagnetic field dependence of D1 for a qualitative anal-ysis. This is because a positive value of the drift co-efficient means that energy is pumped into the dot vi-brations, while a negative value corresponds to damping(cooling) of the vibrations. Therefore, magnetic fieldsfor which D1(A) = 0 and D′

1(A) < 0 correspond to astable stationary state of the dot and a local maximumof the Wigner function. Based on this picture one con-cludes (see Fig. 5) that at low magnetic fields, h < hc1, ashuttling regime with a large vibration amplitude is real-ized, while at high magnetic fields, h > hc1 the situationis more complicated. Here one of two (hc1 < h < hc2;h > hc) or three (hc2 < h < hc) shuttling regimes withdifferent amplitudes can be stable depending on the ini-tial conditions . If the dot is initially in the static state(A = 0) a stable shuttle regime only appears for h < hc

as already mentioned.

Thus the magnetic shuttle device acts in ”opposite”

10

-

-

-

-

-

+

+

+

+

+

+

+

- -

-

h

2A/l

hc

+

G=10

hc1

hc2 10 20

10

20

30

10

FIG. 5: Regions of positive and negative values of the incre-ment coefficient D1(A, h) for Γ = 10. Solid (dashed) linesindicate where the Wigner distribution function for the os-cillation amplitude A has a local maximum (minimum) andhence where the stationary state [D1(A, h) = 0] is stable (un-stable) with respect to small perturbations.

way as compared to electromechanical one. A particu-larly transparent picture of how spintro-mechanics affectsshuttle vibrations emerges in the limit of weak magneticfield H and large electron tunnelling rate ΓS(D) betweendot and source- and drain electrodes. In order to explorethis limit, where ΓS ≫ ω ≫ (µH/h)2/ΓD and ω/2π isthe natural vibration frequency of the dot, we focus firston the total work done by the exchange force F as thedot vibrates under the influence of an elastic force only.In the absence of an external magnetic field the dot is inthis case occupied by a spin-up electron emanating fromthe source electrode. This spin is a constant of motionand hence no electrical current through the device is pos-sible since only spin-down states are available in the drainelectrode. During the oscillatory motion of the dot theexchange force is therefore always directed towards thesource electrode while its magnitude only depends on theposition of the dot, F = F0(x). As a result, no net workis done by the exchange force on the dot. This is becausecontributions are positive or negative depending on thedirection of the dot’s motion and cancel when summedover one oscillation period. A finite amount of work canonly be done if the exchange force deviates from F0(x)as a result of spin flip processes induced by the exter-nal magnetic field. Such a deviation can be viewed asan additional random force FH that acts in the oppositedirection to F0(x). In the limit of large tunneling rate,Γ ≫ µH/h, and small vibration amplitude a spin flipoccurs with a probability ∝ (µH/h)2/(ωΓD) during oneoscillation period and is instantly accompanied by thetunneling of the dot electron into the drain electrode,thereby triggering the force FH . The duration of thisforce is determined by the time δt ∼ 1/ΓS(x(t)) it takesfor the spin of the dot to be“restored”by another electrontunneling from the source electrode.

The spin-flip induced random force FH = −F0(x) isalways directed towards the drain electrode. Hence, its

effect depends on the dot’s direction of motion: as thedot moves away from the source electrode it will be ac-celerated, while as it moves towards the source it will bedecelerated. Since a spin-flip may occur at any point onthe trajectory one needs to average over different spin-flip positions in order to calculate the net work done onthe dot. The result, which depends on the competitionbetween the effect of spin flips that occur at the sameposition but with the dot moving in opposite directions,is nonzero because δt is different in the two cases. As thedot moves away from the source electrode the tunnelingrate to this electrode will decrease while as the dot movestowards the source it will increase. This means that theduration of spin-flip induced acceleration will prevail overthe one for deceleration. As a result, in weak magneticfields, the dot will accelerate with time and one can ex-pect a spintro-mechanical shuttle instability in this limit.

The situation is qualitatively different in the oppositelimit of strong magnetic fields, where Γ ≪ µH/h andthe spin rotation frequency therefore greatly exceeds thetunneling rates. In this case the quick precession of theelectron spin in the dot averages the exchange force tozero if one neglects the small effects of electron tunnel-ing to and from the dot. If one takes corrections due totunnelling into account (having in mind that the sourceelectrode only supplies spin-up electrons) one comes tothe conclusion that the average spin on the dot will be di-rected upwards. This results in a net spintro-mechanicalforce in the direction opposite to that of the net forceoccurring in a weak magnetic field limit. As a result, instrong magnetic fields one expects on the average a de-celeration of the dot. Therefore, there will be no shuttleinstability for such magnetic fields.

As we have discussed above spin-flip assisted electrontunnelling from source to dot to drain in our device re-sults in a magnetic exchange force that attracts the dotto the source electrode. It is interesting to note thatthis is contrary to the effect of the Coulomb force in thesame device. Indeed, since the Coulomb force depends onthe electric charge of the dot it repels the dot from thesource electrode. Hence, while the dot is empty as theresult of a spin-flip assisted tunneling event from dot todrain, an “extra” attractive Coulomb force FQ is active.An analysis fully analogous with our previous analysis ofthe “extra” repulsive magnetic exchange force FH leadsto the conclusion that the effect of the Coulomb forcewill be just the opposite to that of the exchange force.This means that in the Coulomb blockade regime in thelimit of weak magnetic field there is no shuttle instability,while in strong magnetic fields electron shuttling occurs.As was shown the detailed analysis confirms these pre-dictions.

11

V

m

ML

RM

FIG. 6: Single-domain magnetic grains with magnetic mo-ments ML and MR are coupled via a magnetic cluster withmagnetic moment m, the latter being separated from thegrains by insulating layers. The gate electrodes induce anac electric field, concentrated in the insulating regions. Thisfield, by controlling the heights of the tunnel barriers, affectsthe exchange magnetic coupling between different componentsof the system. Reprinted with permission from [64], L. Y.Gorelik et al., Phys. Rev. Lett. 91, 088301 (2003). c© 2003,American Physical Society.

E. Mechanically assisted magnetic coupling

between nanomagnets

The mechanical force caused by the exchange interac-tion represents only one effect of the coupling of magneticand mechanical degrees of freedom in magnetic nano-electromechanical device. A complementary effect is theof mechanical transportation of magnetization, which weare going to discuss in this subsection.

In the magnetic shuttle device presented in Fig. 6, aferromagnetic dot with total magnetic moment m is ableto move between two magnetic leads, which have totalmagnetization ML,R. Such a device was suggested inRef. 64 in order to consider the magnetic coupling be-tween the leads (which in their turn can be small mag-nets or nanomagnets) produced by a ferromagnetic shut-tle. It is worth to point out that the phenomenon weare going to discuss here has nothing to do with trans-ferring electric charge in the device and it is valid alsofor a device made of nonconducting material. The maineffect, which will be in the focus of our attention, is theexchange interaction between the ferromagnetic shuttle(dot) and the magnetic leads. This interaction decaysexponentially when the dot moves away from a lead andhence it is only important when the dot is close to one ofthe leads. During the periodic back-and-forth motion ofthe dot this happens during short time intervals near theturning points of the mechanical motion. An exchangeinteraction between the magnetizations of the dot anda lead results in a rotation of these two magnetizationvectors in such a way that the vector sum is conserved.This is why the result of this rotation can be viewed as atransfer of some magnetization ∆m from one ferromag-net to the other. As a result the magnetization of the dotexperiences some rotation around a certain axis. The to-tal angle φ of the rotation accumulated during the timewhen the dot is magnetically coupled to the lead is an es-

sential parameter which depends on the mechanical andmagnetic characteristics of the device. The continuationof the mechanical motion breaks the magnetic couplingof the dot with the first lead but later, as the dot ap-proaches the other magnetic lead an exchange couplingis established with this second lead with the result thatmagnetization which is “loaded” on the dot from the firstlead is ”transferred” to the this second lead. This is howthe transfer of magnetization from one magnetic lead toanother is induced mechanically. The transfer createsan effective coupling between the magnetizations of thetwo leads. Such a non-equilibrium coupling can be effi-ciently tuned by controlling the mechanics of the shuttledevice. It is particularly interesting that the sign of theresulting magnetic interaction is determined by the signof cos(φ/2). Therefore, the mechanically mediated mag-netic interaction can be changed from ferromagnetic toanti-ferromagnetic by changing the amplitude and thefrequency of mechanical vibrations [64].

IV. RESONANCE SPIN-SCATTERING

EFFECTS. SPIN SHUTTLE AS A ”MOBILE

QUANTUM IMPURITY”.

Many-particle effects add additional dimension to theshuttling phenomena. These effects accompany elec-tronic tunneling between the gate electrodes and themoving nanoisland. The common source of many-particleeffects is the so called ”orthogonality catastrophe” re-lated to multiple creation of electron-hole pairs bothwith parallel and antiparallel spins [65, 66] as a re-sponse of electronic gas in the leads to single electrontunneling. The second-order cotunneling processes un-der strong Coulomb blockade result in effective indirectexchange between the shuttle and the leads. This ex-change is the source of strong scattering and the many-particle reconstruction of the electron ensemble in theleads known as the Kondo effect. Various manifestationsof the Kondo effect in shuttling are reviewed in this sec-tion.The Kondo effect in electron tunneling close to the uni-

tarity limit manifests itself as a sharp zero bias anomalyin the low-temperature tunneling conductance. Many-particle interactions renormalize the electron spectrumenabling ”Abrikosov-Suhl resonances” both for odd [67]and even [68, 69] electron occupations. In the latter casethe resonance is caused by the singlet-triplet crossover inthe ground state (see [70] for a review). In the simplestcase of odd occupancy a cartoon of a quantum well and aschematic Density of States (DoS) is shown in Fig. 7. Forsimplicity we consider a case when the dot is occupied byone electron (as in a SET transistor). The correspond-ing electronic level in the dot is located at an energy −Ed,deep beyond the Fermi level of the leads (ǫF ). The dot isin the Coulomb blockade regime, and the correspondingcharging energy is denoted as EC . The Abrikosov-Suhlresonance [71–73] at ǫF arises due to multiple spin flip

12

scattering, so that the narrow peak in the DoS is re-lated mainly to the spin degrees of freedom (see Fig. 7,upper right panel). The width of this resonance is de-fined by the unique energy scale, the Kondo temperatureTK , which determines all thermodynamic and transportproperties of the SET device through a one-parametricscaling [73]. The Breit - Wigner (BW) width Γ of the dotlevel associated with the tunneling of dot electrons to thecontinuum of levels in the leads, is assumed to be smallerthan the charging energy EC , providing a condition fornearly integer valency regime.

Building on an analogy with the shuttling experimentsof Refs. 74 and 75, let us consider a device where an iso-lated nanomachined island oscillates between two elec-trodes. The applied voltage is assumed low enough sothat the field emission of many electrons, which was themain mechanism of tunneling in those experiments, canbe neglected. We emphasize that the characteristic deBroglie wave length associated with the dot should bemuch shorter than typical displacements allowing thusfor a classical treatment of the mechanical motion of thenano-particle. The condition hω0 ≪ kBTK , necessaryto eliminate decoherence effects, requires for e.g. pla-nar quantum dots with the Kondo temperature TK

>∼100 mK, the condition ω0

<∼ 1 GHz for oscillation fre-quencies to hold; this frequency range is experimentallyfeasible [74, 75]. The shuttling island is then to be con-sidered as a “mobile quantum impurity”, and transportexperiments will detect the influence of mechanical mo-tion on the differential conductance. If the dot is smallenough, then the Coulomb blockade guarantees the singleelectron tunneling or cotunneling regime, which is neces-sary for the realization of the Kondo effect [70, 76].

The above configuration is illustrated in the lowerpanel of Fig. 7: the shuttle of nanoscale size is mountedat the tight string. Its harmonic oscillations are inducedby external elastic force. Unlike the conventional reso-nance case (the resonance level belongs not to the mov-ing shuttle but develops as a many-body peak at theFermi level of the leads. When the shuttle moves be-tween source (S) and drain (D) (see the lower panel ofFig. 7), both the energy Ed and the width Γ acquire atime dependence. This time dependence results in a cou-pling between mechanical, electronic and spin degrees offreedom. If a source-drain voltage Vsd is small enough(eVsd ≪ kBTK) the charge degree of freedom of the shut-tle is frozen out while spin flips play a very important rolein co-tunneling processes. Namely, the Abrikosov-Suhlresonance is viewed as a time-dependent Kondo cloudbuilt up from conduction electrons in the leads dynami-cally screening moving spin localized at the shuttle. Sincethe electrons in the cloud contain information about thesame impurity, they are mutually correlated. Thus, NEMproviding a coupling between mechanical and electronicdegrees of freedom introduces a powerful tool for ma-nipulation and control of the Kondo cloud induced bythe spin scattering and gives a very promising and ef-ficient mechanism for electromechanical transduction on

!" #

Ec

EdεF

TK

!"#$%&'

Γ

FIG. 7: Nanomechanical resonator with spin as a “mobilequantum impurity”.

the nanometer length scale.Cotunneling is accompanied by a change of spin projec-

tion in the process of charging/discharging of the shuttleand therefore is closely related to the spin/charge pump-ing problem [77].A generic Hamiltonian for describing the resonance

spin-scattering effects is given by the same Andersonmodel as above,

H0 =∑

k,α

εkσ,αa†kσ,αakσ,α +

∑

iσ

[Ed − eEx]d†iσdiσ + ECn2

Htunnel =∑

ikσ,α

T (i)α (x)[a†kσ,αdiσ +H.c], (36)

where E is the electric field between the leads. Thetunnelling matrix element depends exponentially on theratio of the time-dependent displacement x(t) and theelectronic tunnelling length λ, see Eq. (15). The time-dependent Kondo Hamiltonian for slowly moving shattlecan be obtained by applying a time-dependent Schrieffer-Wolff transformation [78, 79]:

HK =∑

kασ,k′α′σ′

Jαα′(t)[~σσσ′~S +

1

4δσσ′ ]a†kσ,αak′σ′,α′(37)

where Jα,α′(t) =√Γα(t)Γα′(t)/(πρ0Ed(t)) and ~S =

12d

†σ~σσσ′dσ′ , Γα(t) = 2πρ0|Tα(x(t))|2 are level widths due

to tunneling to the left and right leads.As long as the nano-particle is not subject to an exter-

nal time-dependent electric field, the Kondo temperatureis given by kBT

0K = D0 exp [−(πEC)/(8Γ0)] (for simplic-

ity we assumed that ΓL(0) = ΓR(0) = Γ0; D0 playsthe role of effective bandwidth). As the nano-particlemoves adiabatically, hω0 ≪ Γ0, the decoherence effectsare small provided hω0 ≪ kBT

0K .

Let us first assume a temperature regime T ≫ TK

(weak coupling). In this case we can build a per-

13

0

1

1

0

1

ConductanceG/G

U

Temperature T/T 0K

Ωt

TK(Ω

t)/T

max

K

π

Temperature T/T 0K

FIG. 8: Differential conductance G of a Kondo shuttle forwhich Γ0/EC=0.4. The solid line denotes G for a shuttle withΓL=ΓR, A=λ, the dashed line showsG for a static nano-islandwith ΓL = ΓR, A=0, the dotted line gives G for ΓL/ΓR=0.5,A=0. The inset shows the temporal oscillations (here Ω ≡ ω0)of TK for small A=0.05λ (dotted line) and large A=2.5λ(solid line) shuttling amplitudes. Reprinted with permissionfrom [80], M. N. Kiselev et al., Phys. Rev. B 74, 233403(2006). c© 2006, American Physical Society.

turbation theory controlled by the small parameterρ0J (t) ln[D0/(kBT )] < 1 assuming time as an exter-nal parameter. The series of perturbation theory canbe summed up by means of a renormalization group pro-cedure [73, 79]. As a result, the Kondo temperature be-comes oscillating in time:

kBTK(t) = D(t) exp

[− πEC

8Γ0 cosh(2x(t)/λ)

]. (38)

Neglecting the weak time-dependence of the effectivebandwidth D(t) ≈ D0, we arrive at the following ex-pression for the time-averaged Kondo temperature:

〈TK〉 = T 0K

⟨exp

[πEC

4Γ0

sinh2(x(t)/λ)

1 + 2 sinh2(x(t)/λ)

]⟩. (39)

Here 〈...〉 denotes averaging over the period of themechanical oscillation. The expression (39) acquiresan especially transparent form when the amplitude ofthe mechanical vibrations A is small: A <∼ λ. Inthis case the Kondo temperature can be written as〈TK〉 = T 0

K exp(−2W ), with the Debye-Waller-like ex-ponent W = −πEC〈x2(t)〉)/(8Γ0λ

2), giving rise to theenhancement of the static Kondo temperature.The zero bias anomaly (ZBA) in the tunneling conduc-

tance is given by

G(T ) =3π2

8G0

⟨4ΓL(t)ΓR(t)

(ΓL(t) + ΓR(t))21

[ln(T/TK(t))]2

⟩,(40)

where G0 = e2/h is a unitary conductance. Althoughthe central position of the island is most favorable for

GATE

D

S

L

y

x

z F

B

FIG. 9: Shuttling quantum dot mounted on a moving metallicpendulum. Magnetic field B is applied along z axis. c© 2013,American Physical Society.

the BW resonance (ΓL = ΓR), it corresponds to the min-imal width of the Abrikosov-Suhl resonance. The turningpoints correspond to the maximum of the Kondo temper-ature given by the equation (38) while the system is awayfrom the BW resonance. These two competing effectslead to the effective enhancement of G at high tempera-tures (see Fig. 8).

Summarizing, it was shown in [80] that Kondo shut-tling in a NEM-SET device increases the Kondo temper-ature due to the asymmetry of coupling at the turningpoints compared to at the central position of the island.As a result, the enhancement of the differential conduc-tance in the weak coupling regime can be interpreted asa pre-cursor of strong electron-electron correlations ap-pearing due to formation of the Kondo cloud.

Next we turn to the strong coupling regime, T ≪ TK .We consider this regime for an oscillating cantilever witha nanotip at its end (Fig. 9). Then the motion of ashuttle in y direction is described by the Newton equationwhich we rewrite in a form

y +ω0

Q0y + ω2

0y =1

mF. (41)

where ω0 =√k/m is the oscillator frequency of free can-

tilever, Q0 is the quality factor. F is the Lorentz forceacting on moving cantilever in perpendicular magneticfield

~F = L · ~I × ~B = (0, F, 0). (42)

Here L is the length of the cantilever. ~I is the currentthrough the system.

In this configuration the Kondo cloud induced by spinscattering is formed both in the immovable part of thesetup (drain electrode) and in the oscillating cantilever.

The current ~I subject to a constant source-drain bias Vsd

can be separated in two parts: a dc current associatedwith a time-dependent dc conductance and an ac currentrelated to the periodic motion of the shuttle. While the

14

dc current is mostly responsible for the frequency shift,the ac current gives an access to the dynamics of theKondo cloud and provides information about the kineticsof its formation. In order to evaluate both contributionsto the total current we rotate the electronic states in theleads in such a way that only one combination of thewave functions is coupled to the quantum impurity. Thecotunneling Hamiltonian may be rationalized by means ofthe Glazman-Raikh rotation, parametrized by the angleϑt defined by the relation tanϑt =

√|ΓR(t)/ΓL(t)|.

Both the ac and dc contributions to the current canbe calculated by using Noziere’s Fermi-liquid theory (see[81] for details). The ac contribution, associated with thetime dependence of the Friedel phase δσ [82], is given by

Iac(t) =y(t)

λ

eEC

8Γ0· eVsd

kBTK(t)·tanh

(2[y(t)−y0]

λ

)

cosh2(

2[y(t)−y0]λ

) (43)

(exp(4y0/λ) = ΓR(0)/ΓL(0)). The equation (43) ac-quires a simple form if we assume that the size of Kondocloud RK(y(t)) = hvF /(kBTK(y(t))) where vF is a Fermivelocity. According to Nozieres [81], the Friedel phase δσcan be Taylor-expanded in the vicinity of its resonancevalue δ0σ = π/2 as

δσ(t) =π

2+

eVsdRK(y(t))

hvF+

gµB(σ ·B)RK(y(t))

hvF(44)

and, therefore, d(δ↑+δ↓)/dt ∝ y ·dRK(y)/dy. As a result,

Iac(t) = 2G0Vsdy(t)

vF

dRK(y)

dy. (45)

Thus, the ac current generated in the device due to themechanical motion of the shuttle contains informationabout spatial variation of the Kondo cloud.

The ”ohmic”dc contribution is fully defined by the adi-abatic time-dependence of the Glazman-Raikh angle

IDC(t) = G0Vsd sin2 2ϑt

∑

σ

sin2 δσ (46)

As a result, the ac contribution to the total current canbe considered as a first non-adiabatic correction:

Itot = Iad(y(t))− ydIaddy

hπEC

16Γ0kBT(0)K

(47)

where Iad = 2 · G0 · Vsd cosh−2(2[y(t) − y0]/λ) and T

(0)K

is the Kondo temperature at the equilibrium position.The small correction to the adiabatic current in (47)may be considered as a first term in the expansion overthe small non adiabatic parameter ω0τ ≪ 1, where τis the retardation time associated with the inertia ofthe Kondo cloud. Using such an interpretation one gets

τ = hπEC/(16Γ0kBT(0)K ).

Equation (47) allows one to obtain information about

FIG. 10: Time dependence of the current I0 for differ-ent values of asymmetry parameter u = x0/λ. Here red,blue and black curves correspond to u = 0.5; 1.0; 1.5;.For all three curves shuttle oscillates with amplitudexmax = λ, hω0/(kBT

min

K ) = 10−3, |eVbias|/(kBTmin

K ) =

gµBB/(kBTmin

K ) = 0.1 with T(0)K

= 2K, λ/L = 10−4.Reprinted with permission from [82], M. N. Kiselev et al.

Phys. Rev. Lett. 110, 066804 (2013). c© 2013, AmericanPhysical Society.

the dynamics of the Kondo clouds from an analysis of anexperimental investigation of the mechanical vibrations.The retardation time associated with the dynamics of theKondo cloud is parametrically large compared with thetime of formation of the Kondo cloud τK = h/(kBTK)and can be measured owing to a small deviation fromadiabaticity. Also we would like to emphasize a supersen-sitivity of the quality factor to a change of the equilibriumposition of the shuttle characterized by the parameter u(see Fig. 10). The influence of strong coupling betweenmechanical and electronic degrees of freedom on the me-chanical quality factor has been considered in [82]. Ithas been shown that both suppression Q > Q0 and en-hancement Q < Q0 of the dissipation of nanomechanicalvibrations (depending on external parameters and theequilibrium position of the shuttle) can be stimulated byKondo tunneling. The latter case demonstrates the po-tential for a Kondo induced electromechanical instability.In order to describe these instability, one should discuss

the contribution of ”Kondo force” FK to the right handside part (42) of Eq. (41). This force consists of twocomponents [83]:

FK = − αK + αret

cosh2(y − y0)ω20λ

. (48)

where

αK =πECkBTK(t)

8Γ0λ, (49)

αret = 2yG0VbiasBL tanh(y − y0)τrete−β[1+tanh(y−y0)]/2

Here β = πEC/4Γ0 is the coupling strength of electronicstates. The first term stems from the Kondo cloud adi-abatically following the change of TK(t) induced by themoving shuttle in the source electrode and metallic can-

15

tilever. The second term describes the temporal retarda-tion related to dynamics of Kondo cloud with the char-acteristic time τret = hω0β/(2kBT

minK ). The time depen-

dent Kondo temperature in the strong coupling limit atT ≪ Tmin

K is given by

kBTk(t) = kBTminK exp

β

2[1 + tanh(y(t)− y0)])

.(50)

The kBTminK plays the role of the cutoff energy for Kondo

problem.The instability is controlled by the bias Vbias entering

αret. Fig. 11 illustrates two regimes of Kondo shuttling.Namely, at small bias the Kondo force controlled by ex-ternal fields further damps the oscillator, and we obtainan efficient mechanism of cooling the nano-shuttle. Onthe other hand, at Vbias above some treshold value, thecontribution of the Kondo force enhances the oscillations,and we arrive at the non-linear steady state regime of selfsustained oscillations.Summarizing, we emphasize that the Kondo phe-

nomenon in single electron tunneling gives a verypromising and efficient mechanism for electromechani-cal transduction on a nanometer length scale. Measur-ing the nanomechanical response on Kondo-transport ina nanomechanical single-electron device enables one tostudy the kinetics of the formation of Kondo-screeningand offers a new approach for studying nonequilibriumKondo phenomena. The Kondo effect provides a possi-bility for super high tunability of the mechanical dissi-pation as well as super sensitive detection of mechanicaldisplacement.

V. CONCLUSIONS

During the last several years there has been significantactivity in the study of nanoelectromechanical (NEM)shuttle structures. In this review we concentrate on de-scription of the influence of spin-related effects on thefunctionality of shuttle devices. In particular, we empha-size the importance of electronic spin in shuttle devicesmade of magnetic materials. Spin-dependent exchangeforces can be responsible for a qualitatively new nanome-chanical performance opening a new field of study thatcan be called spintro-mechanics. Electronic many-bodyeffects, appearing beyond the weak tunneling approach,result in single electron shuttling assisted by Kondo-resonance electronic states. The possibility to achieve ahigh sensitivity to coordinate displacement in electrome-chanical transduction along with the possibility to studythe kinetics of the formation of many-body Kondo stateshas also been demonstrated.There are still a number of unexplored shuttling

regimes and systems, which one could focus on in thenearest future. In addition to magnetic shuttle de-vices one could explore hybrid structures where thesource/drain and gate electrodes are hybrids of magnetic

FIG. 11: Panel A: Amplitude dynamics at different values ofthe dimensionless force α (see details in the text). Insets: timetrace of the oscillation at two different fixed point indicated byarrow. Panel B: Saturation amplitude as a function of dimen-sionless force. Different colors denote initial conditions near(black dots) and far (red dots) from the equilibrium positiony0. Insets: amplitude envelope as a function of dimensionlesstime calculated by using Eq. (49). The parameter α variesfrom α = 0 (black) to α = 0.1 (magenta). The equations aresolved for the following set of parameters: β = 8, γ = 10−5,y0 = 0.5 and hω0

kBTmin

K

= 10−3. Reprinted with permission

from [82], T. Song et al., New Journal of Physics, 16, 033043(2014).

and superconducting materials. Then one could expectspintromechanical actions of a supercurrent flow as wellas superconducting proximity effects in the spin dynam-ics in magnetic NEM devices. An additional directionis the study of shuttle operation under microwave radia-tion. In this respect microwave assisted spintromechanicsis of special interest due to the possibility of microwaveradiation to resonantly flip electronic spins. As in ballis-tic point contacts such flips can be confined to particularlocations by the choice of microwave frequency, allowingfor external tuning of the spintromechanical dynamics ofthe shuttle.

VI. ACKNOWLEDGEMENTS

Financial support from the Swedish VR, and the Ko-rean WCU program funded by MEST/NFR (R31-2008-000-10057-0) is gratefully acknowledged. This researchwas supported in part by the Project of Knowledge Inno-vation Program (PKIP) of Chinese Academy of Sciences,Grant No. KJCX2.YW.W10. I. V. K. and A. V. P. ac-knowledge financial support from the National Academyof Science of Ukraine (grant No. 4/13-N). I. V. K. thanksthe Department of Physics at the University of Gothen-burg for hospitality.

16

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