arXiv:cond-mat/0408591v3 [cond-mat.mes-hall] 28 Jul 2005 stedmntaino antchbi nanostruc- hybrid single magnetic context a a this of consisting in of . ture interest demonstration of great charge the of the is development than rather recent manip- A and devices the control with of the deals on ulation which depends functionality [11], spintronics whose is ter been 10]. also have 9, They 8, 7, 5]. [6, 4, [3, theoretically Refs. investigated de- in extensively NEM-SET found by of be confined can studies vices is Experimental cen- motion potential. movable center-of-mass some a whose with island (SET) is tral NEM-SET single- A ob- a a (NEM-SET). experimental of nanoelectromechanical transistor the a result for single-electron is device phenomenon a this suitable of as servation most The systems instability NEMS [2]. nanoelectromechanical certain voltage-induced predicted in bias been nanoelectromechani- recently occur has a to — of mechanism ”shuttle” means cal by transport electron basis the [1]. and (NEMS) nanoelectromechanics systems of nanoelectromechanical mind; scope for in the freedom of is the degrees this designing mechanical by the advantage nanoelec- with an of devices into turned feature dynamically be inherent can change This tronics well operation. may its device spatial during the a Consequently, of forces together. configuration chemical device the the hold with that comparable device become during operation produced distributions inho- charge with mogeneous associated forces Coulomb the scale, molecular h H ag ymaso eketra magnetic external weak a of means by in range vibrations THz excit- nanomechanical an the trigger about to brings opportunity a kind ing in this electrons of shuttled NEM-SET mechanically molecular of ma- to spin possibility the The nipulate [12]. electrodes ferromagnetic two oetwg1 68C ef,TeNetherlands. The Delft, CJ 2628 1, Lorentzweg ∗ rsn drs:KviIsiueo aocec,T Delft, TU Nanoscience, of Institute Kavli address: Present nte ail eeoigbac fcnesdmat- condensed of branch developing rapidly Another — phenomenon nanoelectromechanical pronounced A near the reaches devices electronic of downsizing the As 2 ASnmes 58.j 32.K 57.d 85.35.Be 85.75.-d, 73.23.HK, onse 85.85.+j, the numbers: PACS stable for scenarios Different different Two field. magnetic analyzed. external are tr operation an “shuttle” by and controlled vibrations (NEM-SE be cluster SET to nanoelectromechanical a leading such instability in cluste that metallic shown is nanometer-sized It a containing (SET) sistor eateto ple hsc,Camr nvriyo Tech of University Chalmers Physics, Applied of Department ecnie ffcso h pndge ffedmo h nanome the on freedom of degree spin the of effects consider We 1 eateto hsc,GoeogUiest,S-1 6G¨o 96 SE-412 G¨oteborg University, Physics, of Department .Fedorets D. pnrnc faNneetoehnclShuttle Nanoelectromechanical a of Spintronics C 60 mlcl lcdbetween placed -molecule ∗ , 1 .Y Gorelik, Y. L. Dtd oebr1,2018) 17, November (Dated: 2 .I Shekhter, I. R. otemgei edo h island. the on contribute field not magnetic does which the leads in the to in leads, polarization the spin to the leads. case coupling the symmetric in a magnetization consider the We of direction the to field dicular volt- magnetic field external bias electric The symmetric an A leads. in resulting field. applied, is magnetic age a in spin-degenerate of is absence which has the level, island energy central electron movable single all The a as- lead down. right are the pointing lead in are while left spins up the pointing spins in have electrons to sumed All [13]. leads magnetic devices. NEMS of develop context to the in possibility spintronics interesting value. the critical demonstrates its reaches This leads the between the field onset when electrical the possible for are shuttling scenarios nanoelectromechanical different two of magnitude field the magnetic on the re- magnetic Depending of is external shuttling. that the an instability for sponsible to nanoelectromechanical the sensitivity of strong field this a exploring leads in polarized to results differently between devoted spin-dependent electrons the is of that tunneling Letter show will fields We This anisotropy phenomenon. very magnetic leads. the the than in smaller much field rn nlas( leads elec- in noninteracting describes trons first The terms. several has erator states mentum h lcrn nec edaehl tacntn elec- constant a at held are lead potential each trochemical in electrons The lcrncag and charge electron h ed r ul pnplrzdtela index lead the spin-polarized fully are leads the H h aitna sdt ecieorsystem, our describe to used Hamiltonian The spin-polarized fully with NEM-SET a consider will We npr feetosbtentelascan leads the between electrons of ansport = [ + + upne ewe w antcleads. magnetic two between suspended r D a fsutevbain r found. are vibrations shuttle of t αk † X Uc α,k r sue ob nryidpnet h op- The independent. energy be to assumed are ǫ k eie fti pnrncNEM-SET spintronic this of regimes ooy E429 G¨oteborg, Sweden 96 SE-412 nology, 0 ( )teosto nelectromechanical an of onset the T) a ↑ † − ntelead the in ǫ c 1 αk αk ↑ hnc fasnl-lcrntran- single-electron a of chanics e c n .Jonson M. and rae dsry)a lcrnwt mo- with electron an (destroys) creates ) E ↓ † a α X c αk † ↓ ] = a + X eog Sweden teborg, αk α H > V ,R L, µ + c osc L,R α † α X c α,k ,woeeeto este of densities electron whose ), α ihtecrepnigspin. corresponding the with steba otg.Since voltage. bias the is 0 = − T ∓ 1 α ( gµ ( eV/ X B ) B ,where 2, B/ h a soine perpen- oriented is αk † 2) c h α c E ↑ † + c ewe the between < e ↓ c α † + a sthe is 0 c αk ↓ † α c i ↑ can i (1) 2 also be used as a spin index: L =↑ and R =↓. The ↑ ↑ ↑ ↑ + Γˆ ρ0 Γˆ + Γˆ ρ2 Γˆ + Lγ ρˆ1 , (3) second term represents tunneling of electrons (without q L q L q R q R spin flip) between the island and the leads. The oper- ∂tρ2 = −i [Hosc − xd, ρ2] −{ΓR(x),ρ2} /2 † ator c (cα) creates (destroys) an electron with spin α ↓ ↓ α + Trs Γˆ ρˆ1 Γˆ + Lγ ρ2 . (4) in the dot. The tunneling amplitudes depend exponen- q L q L tially on the displacement X: TL,R(X)= T0 exp{∓X/λ}. Here all lengths are measured in units of the zero point The third and fourth terms describe the single electronic oscillation amplitude x0, all energies in units of ¯hω0 and state in the dot and its coupling to the electric field E −1 time in units of ω0 ; x ≡ X/x0 and p ≡ x0P/¯h are and the magnetic field B. The Zeeman splitting is given dimensionless operators for the oscillator displacement by gµBB, where g is the electronic g-factor and µB is the and momentum correspondingly. Dimensionless electric Bohr magneton. The fifth term represents the Coulomb (d) and magnetic (h) fields are defined by repulsion between two electrons on the island and the 2 2 2 2 last term Hosc = P /2M + Mω0X /2 describes the vi- d ≡ eE/(Mω0x0) , h ≡ gµBB/(¯hω0) . (5) brational degree of freedom associated with the center- of-mass motion of the island. Here M is the mass of the The tunneling of electrons is described by the di- 2 island and ω0 its vibration frequency. For simplicity, we mensionless parameters Γα(x) ≡ 2πDTα(x + d)/(¯hω0), ˆs ˆs ˆs ˆs assume that the temperature is zero. (Γα)s1,s2 ≡ Γα(x)δs1,sδs2,s and Γ+ ≡ ΓL + ΓR. The In the absence of an external magnetic field, electronic damping of vibrations is introduced via the simplest form tunneling through the system described by the Hamilto- of the damping Liouvillian, nian (1) is blocked. Applying a bias voltage and a mag- L •≡−iγ [x, {p, •}] /2 − γ(n +1/2) [x, [x, •]] /2 , (6) netic field allows the electronic transport to be externally γ ω0 manipulated by lifting this “spin-blockade”. By deblock- where γ ≪ 1 is a dimensionless dissipation rate and ing electron tunneling the mechanical degree of freedom βhω¯ 0 nω0 ≡ 1/[e − 1] is the Bose distribution function. is also greatly influenced, the equilibrium position of the One can formally derive Eq. (6) by weakly connecting island becoming unstable if the rate of energy transfer the oscillator to an Ohmic heat bath by adding the terms from the electronic subsystem to the nanooscillator ex- † † Hbath = ¯hωqb bq and Hosc−bath = X gq(b + bq) ceeds a critical value. q q q q to the HamiltonianP (1) and then eliminateP the bath vari- The coupled electronic and mechanical dynamics of the ables in the Born-Markov approximation [15]. dot is governed by a quantum master equation for the The mechanical degree of freedom alone is described by corresponding reduced density operator ρ(t). It can be the density operator ρ+ ≡ ρ0 + Trsρˆ1 + ρ2, which is con- derived from the Liouville-von Neumann equation for the veniently analyzed in the Wigner representation defined total system by projecting out the degrees of freedom by associated with the leads and the thermal bath. The +∞ reduced density operator ρ(t) obtained in this way acts on 1 −ipξ Wρ(x, p) ≡ dξe hx + ξ/2 |ρ| x − ξ/2i (7) the Hilbert space of the dot, which is a tensor product of 2π Z−∞ the Hilbert space of the oscillator and the electronic space of the dot. The latter is spanned by the four basis vectors for a density operator ρ. † † † † |0i, | ↑i ≡ c↑|0i, | ↓i ≡ c↓|0i, and |2i ≡ c↓c↑|0i. In the In an experimentally relevant regime [5], where the electronic basis the operator ρ(t) canbe written as a 4×4 electromechanical coupling parameter matrix whose elements are operators in vibration space. η ≡ d/λ ∼ 1/λ ≪ 1 (8) The diagonal elements ρ0 ≡ h0|ρ|0i and ρ2 ≡ h2|ρ|2i represent the density operators of the empty and doubly occupied oscillator correspondingly. The singly occupied is small, one can find the stationary solution of the sys- tem of Eqs. (2,3,4) perturbatively in terms of the small oscillator is described by the 2 × 2 blockρ ˆ ≡ (ρ)s ,s ≡ 1 1 2 parameters η and γ. (hs |ρ|s i), where s ,s = | ↑i, | ↓i. 1 2 1 2 Rescaling the phase variables X ≡ x/λ, P ≡ p/λ In the high bias-voltage limit (eV >U >> ¯hω) [14] and changing to the polar coordinates X = A sin ϕ, the time evolution of the density operators ρ0,ρ ˆ1 and P = A cos ϕ, one obtains the steady-state equation for ρ2 is determined by the coupled system of dimensionless the Wigner function W+ ≡ Wρ of the oscillator as equations of motion +

∂ϕW+ = {−Γ+(X)C/2+ Lγ } W+ (9)

∂tρ0 = −i [Hosc + xd, ρ0] −{ΓL(x),ρ0} /2 + {η∂P +Γ−(X)C/2} W0−2 − Γ+(X)CW↓−↑/2 . ↓ ↓ + Trs Γˆ ρˆ1 Γˆ + Lγ ρ0 , (2) q R q R Equation (9) couples to the steady state equation for the h vector-function W ≡ [W , W , W , W , W ]T , ∂ ρˆ = −i H − σˆ , ρˆ − Γˆ↓ , ρˆ /2 0−2 ↓−↑ −c +c 0+2 t 1 osc x 1 + 1 where W ≡ W ± W , W ≡ W − W and  2  n o 0±2 ρ0 ρ2 ↓−↑ ρ↓↓ ρ↑↑ 3

15 W±c ≡ Wρ↑↓ ± Wρ↓↑ . Here Γ±(X) ≡ ΓR(X) ± ΓL(X), 2 2 −2 Lγ ≡ γ ∂P P + (1/2λ )∂P and C ≡ cosh iλ ∂P − 1. One can show that in the leading order approximation Shuttle (s) the Wigner function W+(A, ϕ) is ϕ-independent and de- 10 s ) γ termined by Mixed (m) λ

∂AA [f(A)+ D(A) ∂A] W+(A)=0 , (10) d / ( p 5 v where f(A) = (A/2) [γ − ηβ0(A)], Aβ0(A) = Vibronic (v) 2π (0) − 0 (dϕ/π) cos ϕ G0−2(ϕ) ≥ 0 and D(A) > 0 is of sec- (0) 0 ondR order in η and γ. The function G (ϕ) is deter- h 1 0−2 c h mined by the system of differential equations:

(0) (0) FIG. 1: ”Phase diagram” (for Γ+(0) = 0.1) in the electric field D G = Γ (X)G +Γ (X) , (11) ϕ 0−2 − ↓−↑ − (d)- magnetic field (h) plane (see Eq. (5)) showing domains of (0) (0) (0) different nanoelectromechanical behavior of the studied NEM- DϕG = Γ−(X)G − 2hG − Γ+(X) , (12) ↓−↑ 0−2 −c SET device. In the vibronic domain (v) only the vibrational (0) (0) ground state of the device is stable, in the shuttle domain (s) DϕG−c = 2hG↓−↑ ,Dϕ ≡ 2∂ϕ +Γ+(X) (13) the stable state corresponds to developed island vibrations It follows from Eq. (10) that the Wigner function W+ and in the mixed domain (m) both states are locally stable −1 A f(A) with probabilities that are equal along the dashed line p. has the form W+(A) ≈ Z exp − 0 dA D(A) and is n o peaked for amplitudes AM determinedR by the conditions ′ f(AM ) = 0 and f (AM ) > 0. In the vicinity of these am- plitudes W+(A) can be approximated by a narrow Gaus- above the line s. Between these lines (in the m-domain) 2 ′ sian of variance σ = D(AM )/f (AM ). The behavior both states can be stable. The oscillator “bounces” be- of the stationary solution W+(A) is determined by the tween the v− and s-regimes due to random electric forces structure of the positive definite function β0(A), which caused by stochastic variations of the grain charge asso- is bounded, has only one maximum and decreases mono- ciated with tunneling events. tonically for large A. One can show that if hhc it has a minimum there. In the vicinity of A = 0, ω exp(Sv↔s/η), where Sv↔s(d, h) ∼ 1. Since Sv→s 6= 2 0 one finds that β0(A)= γthr/η + O(A ), where Ss→v and η ≪ 1, the difference between the switching rates τ and τ can be exponentially large. This 2Γ h2 1 v→s v←s γ ≡ η 0 , Γ ≡ Γ (0) . (14) implies that the probabilities for the system to be in one thr 1+Γ2 h2 +Γ2 0 2 + 0 0 or the other of the two regimes can be very different. From this equation it follows that the electromechani- The line p in Fig. 1 corresponds to Sv→s = Sv←s and cal properties for a low-transparency junction (with a therefore to equal rates for the transitions v → s and resistance in the Gohm range) is sensitive to very weak s → v. Below this line, the probability for the system magnetic fields of order 1-10 Oe. Such weak fields have to be in the v-regime is exponentially larger than for it a negligible effect on the internal magnetization of the to be in the s-regime, while above the dominance of the leads, which is why this effect was not considered here. s-regime is exponentially large. Due to the smallness of The presence of various stationary regimes can be il- the electromechanical coupling, η ≪ 1, the transition be- lustrated by a “phase diagram” in the (d, h)-plane. Fig- tween the two regimes is very sharp. Hence the change ure 1 shows three domains that correspond to three dif- of vibration regime can be regarded as a “phase transi- ferent types of behavior of the nanomechanical oscillator. tion”. Such a transition will manifest itself if the varia- In the “vibronic” domain (v), defined by the condition tion of the external fields is adiabatic on the time scale η/γ < 1/ [max β0(A)] (h), the NEM-SET system is stable of max{τs↔v}. One can expect enhanced low-frequency < −1 with respect to mechanical displacements of the island noise, ω ∼ τs↔v, around the line p as a hallmark of the from its equilibrium position. The “shuttle” domain (s) transition. is determined by the condition γ<γthr and corresponds In the opposite non-adiabatic limit, either the s- or the to developed island vibrations of a classical nature. The v-regime is “frozen in” in the mixed domain after cross- third domain is the “mixed” domain (m). It appears be- ing the line p. Thus if one starts in the v-domain the cause the v- and s-regimes become unstable at different v-regime persists until the system crosses the line s, and combinations of electric (d) and magnetic (h) fields if h if one starts from the s-domain the s-regime persists until exceeds a critical value hc (cf. [9, 16]). the system crosses the line v. Hence, one observes a hys- Hence, while the shuttle regime is unstable below the teretic behavior of the non-adiabatic shuttle transition. line v in Fig. 1, the vibronic regime becomes unstable As one can see from Fig. 1, there are two different sce- 4 narios for the onset of shuttle vibrations. If one crosses (NEM-SET) system consisting of a metallic island sus- over from the v- to the s-domain when hhc, on the other hand, the onset is hard. In this gree of freedom of the island by an external magnetic case the vibration amplitude shows a step at the transi- field. Different stable operating regimes of the magnetic tion point (Fig. 3), which corresponds to crossing either NEM-SET were found and transitions between them in- the p- or the s-line depending on whether the transition duced by varying the electric and magnetic fields were is adiabatic or not. analyzed. We have hence demonstrated that magnetic- field-controlled spin effects can lead to a very rich behav- 2.5 ior of nanomechanical systems.

2 Although we have considered fully spin-polarized leads and assumed a symmetric set-up, the overall qualitative 1.5 picture does not change if these conditions are relaxed. λ 0.4 /

C A partial polarization, allowing a shuttle instability even ) A 1 0 at zero external magnetic field, makes the system sensi- tive to the magnetic field in a more narrow interval of 0.5 I / (e f electric fields. An asymmetric coupling to the leads in- 0 h 0.3 duces an additional magnetic field along the direction of 0 0.1 0.2 0.3 0.4 0.5 their spin-polarization. This field reduces the rate of en- h ergy pumping into the oscillator, moving the domains in Fig. 1 to higher values of electric and magnetic fields. FIG. 2: Steady-state amplitude AC of shuttle vibrations and current I through the NEM-SET device for parameters cor- This work was supported in parts by the Swedish responding to a ”soft” transition between the vibronic and Foundation for Strategic Research, by the Swedish Re- shuttle domains of Fig. 1 (Γ+(0) = 0.1 and d/(γλ) = 14.3). search Council and by the European Commission through project FP6-003673 CANEL of the IST Priority. The views expressed in this publication are those of the au- 2.5 thors and do not necessarily reflect the official European Commission’s view on the subject. 2

λ 1.5 0.4 / C ) A 1 0 [1] M. L. Roukes, Physics World 14, 25 (2001); A. N. Cle- land, Foundations of Nanomechanics (Springer) 2003; M. I / (e f 0.5 Blencowe, Phys. Rep. 395, 159 (2004) [2] L. Y. Gorelik et al., Phys. Rev. Lett. 80, 4526 (1998). 0 0.3 h [3] A. Erbe et al., Phys. Rev. Lett. 87, 096106 (2001). 0 0.1 0.2 0.3 0.4 0.5 [4] D. V. Scheible and R. H. Blick, Appl. Phys. Lett. 84, h 4362 (2004). [5] H. Park et al., Nature 407, 57 (2000). FIG. 3: Hysteretic behavior of the steady-state shuttling am- [6] R. I. Shekhter et al., J. Phys. C 15, R441 (2003). plitude AC and current I through the NEM-SET device for [7] D. Fedorets, Phys. Rev. B 68, 033106 (2003). parameters corresponding to a ”hard” transition between the [8] D. Fedorets et al., Phys. Rev. Lett. 92, 166801 (2004). vibronic and shuttle domains of Fig. 1 (Γ+(0) = 0.1 and [9] T. Novotny, A. Donarini, and A.-P. Jauho, Phys. Rev. d/(γλ) = 11.1). Lett. 90, 256801 (2003); T. Novotny et al., Phys. Rev. Lett. 92, 248302 (2004). [10] A. Y. Smirnov, L. G. Mourokh, and N. J. M. Horing, One can show that the expression for the steady state Phys. Rev. B 69, 155310 (2004). current through the system is [11] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 75, 715 (2004); Y. Tserkovnyak, cond-mat/0409242. [12] A.N. Pasupathy et al., Science 306, 86 (2004). I = e dXdP ΓL(X)[W+ + W0−2 + W↓−↑]/2 . (15) Z Z [13] Materials with almost 100% are of great current interest, see Colossal Magnetoresistive Oxides, Typical plots of the current in the cases of soft and hard Advances in Condensed Matter Science Vol. 2, edited by transitions are shown in Figs. 2 and 3, respectively. Y. Tokura (Gordon and Breach, Amsterdam, 2000) In conclusion, we have considered ”shuttle” phenom- [14] Voltages at which the shuttle phenomenon should occur ena in a nanoelectromechanical single-electron transistor in realistic systems are large compared to the Coulomb 5

energy U ∼ 1 eV [12]. Singapore, 1999), 2nd ed. [15] U. Weiss, Quantum Dissipative Systems, Series in Mod- [16] A. Isacsson et al., Physica B 255, 150 (1998). ern Vol. 10 (World Scientific,