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arXiv:cond-mat/0104200v3 [cond-mat.mes-hall] 26 Apr 2002

Europhysics Letters

PREPRINT

Vibrational instability due to coherent tunneling of elec-

trons

D. Fedorets(∗), L.Y. Gorelik, R.I. Shekhter and M. Jonson

Department of Applied Physics, Chalmers University of Technology and G¨ oteborg Uni-

versity - SE-412 96 G¨ oteborg, Sweden

PACS. 73.63.-b – Electronic transport in mesoscopic or nanoscale materials and structures..

PACS. 73.23.Hk – Coulomb blockade; single-electron tunneling.

PACS. 85.85.+j – Micro- and nano-electromechanical systems (MEMS/NEMS) and devices.

Abstract. – Effects of a coupling between the mechanical vibrations of a quantum dot placed

between the two leads of a single electron transistor and coherent tunneling of electrons through

a single level in the dot has been studied. We have found that for bias voltages exceeding a

certain critical value a dynamical instability occurs and mechanical vibrations of the dot develop

into a stable limit cycle. The current-voltage characteristics for such a transistor were calculated

and they seem to be in a reasonably good agreement with recent experimental results for the

single C60-molecule transistor by Park et al. ( Nature 407, (2000) 57).

Introduction. –

densed matter physics. A coupling between strongly pronounced mesoscopic features of the

electronic degrees of freedom (such as quantum coherence and quantum correlations) and de-

grees of freedom connected to deformations of the material produces strong electromechanical

effects on the nanometer scale. The mesoscopic force oscillations in nanowires [3–5] observed a

few years ago is an example of such a phenomenon. Investigations of artificially-made nanome-

chanical devices, where the interplay between single-electron tunneling and a local mechanical

degree of freedom significantly controls the electronic transport, is another line of nanoelec-

tromechanics [6–15]. For one of the nanomechanical systems of this kind, the self-assembled

single-electron transistor, a new electromechanical phenomena - the shuttle instability and

a new so-called shuttle mechanism of the charge transport were recently predicted in [12].

It was shown that a small metallic grain attached to two metallic electrodes by elastically

deformable links breaks into oscillations if a large enough bias voltage is applied between the

leads. For the model system studied in [12], it was also shown that a finite friction is required

for the oscillation amplitude to saturate and for a stable regime of oscillations to develop.

An essential assumption made in [12] is that the relaxation mechanisms present are strong

enough to keep the electron systems in each of the conducting parts of the transistor in local

equilibrium (as assumed in the standard theory of Coulomb blockade [16,17]). Such relaxation,

which destroys any phase coherence between electron tunneling events, allows a description of

Nanoelectromechanics [1,2] is a new, quickly developing field in con-

(∗) E-mail: dima@fy.chalmers.se

c ? EDP Sciences

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EUROPHYSICS LETTERS

x

µL = eV

µR = 0

LeadLead

Dot

Fig. 1 – Model system consisting of a movable quantum dot placed between two leads. An effective

elastic force acting on the dot from the leads is described by the parabolic potential. Only one single

electron state is available in the dot and the non-interacting electrons in each have a constant density

of states.

the whole electronic kinetics by means of a master equation for the occupation probabilities of

the dot. It is clear that such an incoherent approach must fail if the size of the grain is small

enough. Firstly, a decrease of the grain size results in a decrease of its mass and consequently

in an increase of the frequency of the mechanical vibrations, which for nanometer-size clusters

is of the order of 0.1 ÷ 1THz and might exceed the electron relaxation rate in the grain.

Secondly, the electron energy level spacing in a nanometer-size grain δ ∼ ǫF/N, (N is the

number of electrons) can be of the order of 10K and exceed the operational temperatures. In

this situation the discreteness of the energy spectrum can be very important. What is more,

a possibly strong tunneling-induced coupling between electronic states in the grain and in the

leads, which results in large quantum fluctuations of the charge in the grain, must be included.

For all of the above reasons a new approach is needed for a description of the electron

transport through a nanometer-size movable cluster or quantum dot. The non-trivial question

then arises whether or not the coherent electron tunneling through a movable dot causes any

electromechanical instability. Such a question is of notable practical significance in view of the

recent experiment by Park et al. [7], where the current through so-called single-C60-molecule

transistors was measured and anomalies — including a few equidistant step-like features — in

the I-V characteristics observed. This current steps were interpreted in [7] as a manifestation of

the coupling between electron tunneling and the center-of-mass vibrational degree of freedom

of the molecule.

Theoretical model. –

we consider a model system consisting of a movable quantum dot placed between two bulk

leads. An effective elastic force acting on the dot due to interaction with the leads is de-

scribed by the parabolic potential presented in fig. 1. We assume that only one single electron

state is available in the dot and that the electrons in each lead are non-interacting with a

constant density of states. At the same time we treat the motion of the grain classically. The

Hamiltonian for the electronic part of the system is

To investigate the influence of the above electromechanical coupling

H =

?

α,k

(ǫαk− µα)a†

αkaαk+ ǫd(t)c†c +

?

α,k

Tα(t)(a†

αkc + c†aαk). (1)

Here TL,R= τL,Rexp{∓x(t)/λ} is the position-dependent tunneling matrix element, ǫd(t) =

ǫ0− Ex(t) is the energy level in the dot shifted due to the voltage induced electric field

E/e = χV , χ is a parameter characterizing the strength of the electrical field as a function

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D. Fedorets et al.: Vibrational instability due to tunneling

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of the bias voltage V , e < 0 is the electron charge, x measures the displacement of the dot,

a†

αkcreates an electron with momentum k in the corresponding lead, α = L,R is the lead

index, c†creates an electron in the dot and λ is the characteristic tunneling length [12]. The

first term in the Hamiltonian describes the electrons in the leads, the second — the movable

quantum dot and the last term — tunneling between the leads and the dot.

The evolution of the electronic subsystem is determined by the Liouville-von Neumann

equation for the statistical operator ˆ ρ(t),

i∂tˆ ρ(t) = [ˆH, ˆ ρ(t)]−,(2)

while the center of mass motion of the dot is governed by Newton’s equation,

¨ x + w2

0x = F/M . (3)

Here w0=

the harmonic potential, F = − < ∂ˆH/∂x > and < • >= Tr{ˆ ρ(t)•}. The force F on the RHS

of eq. (3) is due to the coupling to the electronic subsystem and consists of two terms,

?k/M, M is the mass of the grain, k is a constant characterizing the strength of

F(t) = −iEG<(t,t) + 2λ−1?

αk(t,t′) ≡ i?a†

α,k

(−1)αTα(t)Im?G<

αk(t′)c(t)? are the lesser Green functions, and

αk(t,t)?, (4)

where G<(t,t′) ≡ i?c†(t′)c(t)?, G<

α = 0(1) for the left (right) lead. The first term describes the electric force that acts on the

charge in the dot; the second term is an exchange force, which appears due to the position

dependence of the tunneling matrix elements TL,R. The force F depends on the correlation

functions G<(t,t) and G<

αk(t,t), which can be computed exactly in the wide-band limit (ρα=

const) by using the Keldysh formalism [18].

Following the standard analysis [19] we express the correlation function G<

of the Green functions of the dot,

αk(t,t) in terms

G<

αk(t,t′) =

?

dt1Tα(t1)?Gr(t,t1)g<

αk(t1,t′) + G<(t,t1)ga

αk(t1,t′)?. (5)

Here Gris the retarded Green function of the dot and ga(<)

tion of the leads for the uncoupled system. The Dyson equation for the retarded (advanced)

Green function has the following form: Gr(a)= gr(a)+ gr(a)Σr(a)Gr(a), where Σr(a)(t1,t2) =

?

G<= GrΣ<Ga, where Σ<(t1,t2) =?

?

where

?t

Γ(t) = 2π?

tunneling matrix elements.

An important question is now whether or not the mechanical stability of the transistor

configuration is affected by coherently tunneling electrons.

αk

is the advanced (lesser) Green func-

α,kTα(t1)gr(a)

equation can be solved exactly. The lesser Green function G<is given by the Keldysh equation

α,kTα(t1)g<

As a result we obtain a general expression for the force F of the form

α,k(t1,t2)Tα(t2). In the wide-band limit Σr(a)(t1,t2) ∝ δ(t1−t2) and this Dyson

αk(t1,t2)Tα(t2).

F(t) =

α

ρα

?

dǫfα(ǫ)

?

E |Bα(ǫ,t)|2+ 2(−1)α

λ

Tα(t)Re[Bα(ǫ,t)]

?

, (6)

Bα(ǫ,t) = −i

−∞

dt1Tα(t1)exp

?

i

?t

t1

dt2

?

ǫ − ǫd(t2) + iΓ(t2)

2

??

,

αραT2

α(t), ρα is the density of states in the corresponding lead and fα(ǫ) =

[exp(β(ǫ −µα)) + 1]−1. It is worth mentioning that eq. (6) is valid for arbitrary values of the

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Instability.

dot we expand the RHS of eq. (3) to first order with respect to the displacement: F(t) =

?t

w2− (w2

1) = −Dw/M ,

–

In order to investigate the stability of the equilibrium position of the

−∞dt′D(t − t′)x(t′). After a Fourier transformation of the obtained equation we get the

following dispersion equation for the frequency:

0− w2

(7)

where

Dw= −

?

dǫ

2π

?

α

Γαfα{RαG+

w+ R∗

αG−

−w},

Rα=

?????EG+

0+(−1)α

λ

????

2

− ig

?

E |G0|2+ G+

0

(−1)α

λ

??

,

w2

and g = (ΓL− ΓR)/λ.

The criterion for instability is that the frequency ω has a positive imaginary part. When

Γ/(Mw2

we obtain that Im[w] ≈ −Im[Dw0]/(Mw0). From now on we consider only the case of weak

electromechanical coupling. An analytical analysis can not be performed in the general case,

but for a large symmetrically applied bias voltage and zero temperature one can show that

Im[Dw0] is negative:

1=?

αΓα

?

dǫfαRe[G+

0]/(πλ2M), Γα= 2πρατ2

α, Γ =?

αΓα, G±

w(ǫ) = [ǫ+w−ǫ0±iΓ/2]−1

0λ2) ≪ 1 and Eλ/(?w2

0+ Γ2) ≪ 1 (the case of weak electromechanical coupling)

Im[Dw0] ≍ −

4Ew0

0+ Γ2)λ(w2

ΓLΓR

Γ

. (8)

The instability that follows from eq. (8) is in contrast with the behavior at low voltages

where it can be shown that Im[Dw0] is positive. Therefore, a finite threshold voltage for

the instability exists in the system. A simple expression for the threshold voltage can be

found for a symmetric junction in the case of weak tunneling under the above conditions:

eVc= 2(ǫ0+ w0).

Under the condition of weak electromechanical coupling, the displacement x(t) of the dot

can be represented in a harmonic oscillation form A(t)cos(wt + φ(t)) with an amplitude A(t)

and a phase φ(t) that slowly vary on the scale of w−1

oscillations one can get the following equation for the evolution of the amplitude (see for

example [12]):

dA2

dtMπw0

Here W(A2) =

?

oscillations corresponds to W = 0 and dW(A2)/dA2< 0. Typical W(A2)-curves are depicted

in fig 2 for the parameters taken from the experiment described in [7]. When an applied

voltage is lower than the threshold voltage the work W is negative for all amplitudes. This

implies that the equilibrium position of the grain is stable. When the voltage is higher than

the threshold voltage the function W(A2) is positive for 0 < A < Ac(V ) (which corresponds

to a slow increase of the oscillation amplitude), negative for A > Ac(V ) (which corresponds

to a slow decrease of the amplitude) and equal to zero at A = Ac(V ). This means that

when the applied voltage exceeds the threshold value the amplitude of the oscillation slowly

increases until it develops into a stable limit cycle with amplitude Ac(V ). As one can see from

fig. 2, the amplitude of the limit cycle is of the order of λ ∼ 0.1nm and is considerably larger

than the amplitude of zero-point oscillations of the grain, which for the C60molecule in the

experiment [7] is approximately 3pm.

0. By averaging over the fast harmonic

=W(A2)

. (9)

dxF is the work done by the force F on the dot during one period of

oscillation with a constant amplitude A. We conclude from eq. (9) that the stable regime of

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D. Fedorets et al.: Vibrational instability due to tunneling

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0 0.20.4 0.60.8

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

(A / λ)2

W / Γ

11 meV

16 meV

21 meV

26 meV

Fig. 2 – The energy W pumped into the mechanical degree of freedom during one period of oscillation

for different values of the applied voltage V : w0 = 5meV, T = 0.13meV, ǫ0 = 6meV, Γ = 1µeV and

ΓR/ΓL = 9.

Current. –

vibration-induced inelastic tunneling of electrons can significantly contribute to the current.

The time-averaged current through the system in the stable regime where oscillations of the

limit cycle amplitude Ac(V ) and frequency w0 =

form:

In the regime where the stable limit cycle oscillations have developed the

?k/M have developed has the following

Bα

τα

I = −eΓL

2πT

?T

0

dt

?

dǫ

?

e−2x

λ

?

α

Γαfα

????

????

2

+ 2e−x

λfLIm

?BL

τL

??

, (10)

where x(t) = Accos(w0t) and T = 2π/w0is the period of oscillations. This expression is valid

under the same general conditions as eq. (6).

Under the condition that the effective level broadening˜Γ ≪ w0(where˜Γ = ΓJ0(i2Ac/λ))

eq. (10) can be simplified to

I ≈eΓLΓR

Γ

+∞

?

m=−∞

{fL,mξ2m

L

− fR,mξ2m

R}J2

m(Acη), (11)

where fα,m= fα(ǫ0+ mw0), η =?(E/w0)2− λ−2, ξα= −[(E/w0) + (−1)α/λ]/η and Jmare

that voltage has been applied only to the left lead (according to the experiment [7]). The

main characteristic feature of all obtained I-V curves is that they show only a few equidistant

steps which are followed by step-less behavior of the curves. The distance between the steps is

given by the vibrational frequency w0and their heights can vary depending on the parameters.

The steps following the first one is due to the development of a vibrational instability and a

transition into the associated charge transfer regime. The obtained behavior of the I-V-curves

is in reasonably good agreement with the experimental data [7]. Best fit to the published

experimental I-V-curves is obtained for an asymmetric coupling to the leads (ΓR/ΓL≈ 9).

When the ratio ΓR/ΓLdecreases both the vibration-induced current jumps and a high-voltage

slope of the I-V-curve increase deviating from the experimental data.

An alternative theoretical description of the experiment based on a photon-assistedtunneling-

like picture [15] also shows reasonable agreement with the experimental data [7]. Therefore it

Bessel functions of the first kind. The typical I-V curves are shown in fig. 3, where we assume

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EUROPHYSICS LETTERS

0102030405060

0

0.05

0.1

0.15

eV, meV

I, nA

χ = 0.20 nm−1

χ = 0.15 nm−1

χ = 0.10 nm−1

Fig. 3 – I-V curves for different values of the parameter χ which characterizes the strength of the

electric field between the leads for a given voltage: w0 = 5meV, T = 0.13meV, ǫ0 = 6meV,

and Γ = 2.3µeV. Best fit to [7] is obtained for an asymmetric coupling to the leads; here we use

ΓR/ΓL = 9.

is difficult to conclude with certainty that it is the above instability which is responsible for

the specific features observed in the experiment. Further experiments are needed to clarify

the picture. Such experiments may involve an investigation of the charge fluctuation on the

grain or of current noise.

Conclusion. –

electrons through a single quantized energy level in the central island — or dot — of a single

electron transistor and vibrations of the dot. We have found that for bias voltages exceeding

a certain critical value a dynamical instability occurs and mechanical vibrations of the center

of mass of the dot develop into a stable limit cycle. The effect of this vibrations on the current

through the system were also studied. I-V characteristics calculated in our model were found

to be in a reasonably good agreement with recent experimental results of Park et al. [7] for

the single C60-molecule transistor.

We have studied the effect of a coupling between coherent tunneling of

∗ ∗ ∗

Financial support from the Swedish SSF through the QDNS program (D. F., L. G.) and

the Swedish NFR/VR (R. S.) and the U.S. Department of Energy Office of Science through

contract No. W-31-109-ENG-38 is gratefully acknowledged.

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