Electromechanics of suspended nanowires

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E

LECTROMECHANICS OF SUSPENDED

NANOWIRES

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E

LECTROMECHANICS OF SUSPENDED

NANOWIRES

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magniﬁcus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 15 oktober 2012 om 15:00 uur

door

Giorgi L

ABADZE

Master of Physics,

Ivane Javakhishvili Tbilisi State University, Georgië, geboren te Tbilisi, Georgië.

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Dit concept-proefschrift is goedgekeurd door de promotor: Prof. dr. ir. G. E. W. Bauer

Copromotor: Dr. Y. M. Blanter Samenstelling promotiecommissie:

Rector Magniﬁcus, voorzitter

Prof. dr. ir. G. E. W. Bauer Technische Universiteit Delft, promotor Dr. Y. M. Blanter Technische Universiteit Delft, copromotor Prof. dr. ir. H. S. J. van der Zant Technische Universiteit Delft

Prof. dr. ir. T. M. Klapwijk Technische Universiteit Delft Prof. dr. G. I. Japaridze Ilia state University, Georgië Prof. dr. F. Pistolesi Universite Bordeaux 1, Frankrijk Dr. J. M. Thijssen Technische Universiteit Delft

Prof. dr. O. Yarovyi Technische Universiteit Delft (reservelid)

Keywords: NEMS, dissipation, mode coupling, parametric reso-nance.

Cover design: G&K

Casimir PhD Series, Delft-Leiden 2012-27 ISBN 978-90-8593-134-8

An electronic version of this dissertation is available at http://repository.tudelft.nl/.

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Research is what I’m doing when I don’t know what I’m doing. Wernher Von Braun

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1

Introduction

T

he two questions ”how” and ”why” drive the progress of our civilization. One of these questions drove an artisani back in late8t h _{century to}

gen-erate a glittering effect on the surface of pots[2]. He/She did not know why adding copper and silver salts and oxides together with vinegar or ochre on the surface of glazed pottery after heating was giving shiny and colorful effect, but he/she found the answer to the question how to make his handmade pottery more attractive for customers. The question why this effect occurred was only answered a millennium later by Michael Faraday. The cause of this optical effect were the ‘minute globular portions’ of the used metal (gold or silver), forming the outer layers of the glaze. Faraday provided the first description of the optical properties of these metallic particles[3], that later were called nanoparticles.

Probably, the same two questions motivated Richard Feynman to give the famous talk ”There’s Plenty of Room at the Bottom” at the annual meeting of the American Physics Society[4]. He provided a vision of molecular machines building complex materials with atomic precision as well as storing information in miniaturized volumes, using the example of writing ‘the entire 24 volumes of the Encyclopedia Britannica on the head of a pin’. He tried to provoke every interested person to think about new ways of fabrication by proposing a competition on two challenges. He oﬀered a $1000 prize to ‘the ﬁrst guy who can take the information on the page of book and put it on the area

1/25000 smaller in linear scale in such manner that it can be read by an

i_{An artisan is a person who makes items that may be functional or strictly decorative[}₁_].

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2 1. Introduction

electron microscope’ and another $1000 prize to ‘the first guy who makes an operating electric motor-a rotating electric motor which can be controlled from the outside and, not counting the lead-in wires, is only 1/64inch cube’. Even though Feynman’s second challenge was accomplished in a year without any novel methods of fabrication, his talk started a whole new field. Since then many new discoveries and developments were made on the way of the fabrication of the ever-smaller artificial devices.

Typical motivations for miniaturization include the convenience of de-creased size, smaller production cost per unit (less material is needed, so fabri-cation is cheaper), better device performance (lower energy consumption and faster operation). Decreasing of the size of the devices also leads to the change of their properties and can add new functionalities. Two mechanisms that bring such changes are:

• Finite size effects. Reducing the size of a structure eventually results in reduced dimensionality. Depending on the dimensionality of the sys-tem, the same governing laws often manifest themselves in qualitatively different behavior that can have important consequences. There are sev-eral phenomena that do not occur in three-dimensional (3D) bulk metals but can be observed in lower dimensions. For example, quantization of the conductance of a quantum point contact, Coulomb blockade and single-electron charging effects in zero-dimensional conductors [5]; Spin-charge separation and Luttinger-liquid behavior in a one-dimensional conduction channel[6, 7]; Integer and fractional quantum Hall effect in two-dimensional conductors [8,9].

• Classical vs. quantum effects. It is known that if on a macroscopic level classical mechanics gives a proper description of a system and governing laws, then quantum mechanics is required on a microscopic level. There-fore shrinking the device sizes bring us to the crossover from classical to quantum mechanics. This results in the fundamentally new phenomena. Universal conductance fluctuations [10,11] and the Aharonov-Bohm ef-fect [12] are two examples of such quantum effects.

Nanomechanical systems (NMS) are an essential part of the miniaturized world. These devices integrate electrical, optical and mechanical function-ality on the nanoscale. NMS combine small mass (as small as zeptograms,

10−21g), high fundamental frequencies (up to gigahertz, 109H z) of the me-chanical resonator, low power consumption and high surface area to volume ratio. Therefore they are interesting for applications in sensing, portable power generation, energy harvesting, drug delivery and imaging [13–21]. There are

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3

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many applications that are available as research prototypes, such as: nano-electromechanical sensors[22], carbon nanotube single electron transistors[23], relays [24] and switches with nanotubes[25, 26], ultrasharp tips for atomic force microscopy (e.g., single-walled carbon nanotubes mounted on the tip of an atomic force microscopy cantilever)[27,28], etc.

One of the main reasons why nanomechanical systems have attracted the attention of the scientific world is that they can be cooled to the temperatures so low that the resonator will be nearly always in its quantum-mechanical ground state. So these systems are ideal candidates for studying quantum be-havior of macroscopic objects. Developments of sensitive detection techniques, capable of detecting zero point motion of nanoresonators, encouraged many research groups to take part in the race of the demonstration of the quantum limit of motion. Recently the quantum mechanical behavior of mechanical resonators was detected in several experiments: the groups of Cleland and Martinis observed ground state of a 6 GHz mechanical resonator[29]. The resonator was cooled to the dilution refrigerator temperatures. To read out the zero point motion of the resonator, it was coupled to a superconducting flux qubit. Large coupling between the two quantum systems was achieved by using the piezoelectric properties of the mechanical resonator material. The flux qubit, acting as a two-level quantum system, has a different response de-pending on the occupation of the mechanical resonator. By measuring the flux qubit response it was demonstrated that the mechanical system can be cooled to its ground state as well as excited with individual mechanical quanta. Completely different design was used by Teufel et. al. [30] to demonstrate a quantum mechanical ground state of a mechanical drum resonator. The drum resonator was integrated in a superconducting microwave cavity which pro-vided strong phonon-photon coupling and was used as a cooling mechanism at the same time. In the experiment performed by Painter and his coworkers [31], a cavity optomechanical system was used to observe the quantum motion of a patterned silicon nanobeam, which formed an optomechanical crystal ca-pable of localizing both optical and acoustic waves. The in-plain mechanical breathing mode of the nanobeam at ωm= 3.99GHz was coupled to two

spa-tial optical modes. One spaspa-tial mode was used to cool the mechanical mode via radiation pressure damping. The other cavity mode provided read out of mechanical motion. By measuring the asymmetry in the motional side-bands of the spectra of the readout signal it was shown that the mechanical oscillator was cooled down close to its ground state. Optomechanical system with diﬀerent design was used by Kippenberg and his coworkers to observe quantum-coherent coupling of the mechanical oscillator to an optical cavity mode [32]. In this experiment ’quantum coherent’ optomechanical coupling

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4 1. Introduction

was achieved - that is, when the coherent coupling exceeds both the optical and the mechanical decoherence rates. The authors reported full control of the quantum state of a mechanical oscillator with optical ﬁeld. These experiments represent a breakthrough and show that quantum behavior can be encoded in the motion of a mechanical resonator. They open the door of a new exciting research ﬁeld.

1.1 Experimental realization of mechanical

sys-tems

T

here are two approaches to fabrication nanomechanical systems: top-down and bottom-up. The top-down approach uses traditional microfabrication methods, such as optical and electron beam lithography, to manufacture the devices. This method of fabrication allows a large degree of control over the resulting structures but the resolution is limited. The masses of the devices fabricated by the top-down approach range from attograms to kilograms, and resonance frequencies range from a Hz to a GHz. Decreasing the size of these devices results in lower values of the quality factor, a measure for the energy dissipation in the system. The surface defects introduced during fabrication provide channels for dissipation, which become the main sources of the energy loss as the surface-to-volume ratio increases [33]. Thus smaller devices fabri-cated by a top-down approach are expected to have a low quality factor. Low quality factor also means an undesirable short coherence time of the quantum states of these systems. In bottom-up approach the concepts of molecular self-assembly and molecular recognition is used to self-organize useful conﬁgu-rations. This allows the fabrication of smaller structures, but often at the cost of limited control of the fabrication process. Bottom-up devices are expected to be defect-free so they should not suﬀer from excessive surface losses. Exam-ples are inorganic nanowires, carbon nanotubes and few-layer graphene. Using these bottom-up materials, one can make defect-free mechanical devices with true nanometer dimensions. High Q-values are therefore expected for these devices which, combined with their low masses, make them ideal to study the quantum behavior of nanomechanical systems.

Here we would like to discuss the basic features of several devices that use nanomechanical systems as building blocks.

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1.1. Experimental realization of mechanical systems 5 { {

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Input laser Cavity Mechanichal oscillator x Detector

Figure 1.1: Schematic picture of a Fabry-Perot cavity coupled to an oscillator.

1.1.1 Cavity based realisations

Modern cavity-mechanical systems, in which the mechanical oscillator is cou-pled to an electromagnetic field of a cavity, can be found at different scales and geometries, from multi-kilometer long gravitational-wave detectors to sub-micrometer superconducting microwave circuits. Probably the hottest field of the last decade is optomechanics. In optomechanical systems mechani-cal degrees of freedom are coupled to an electromagnetic interferometer or cavity[34,35]. Here we would like to discuss the basic features of the Fabry-Perot cavity shown in Fig. 1.1. It consists of two mirrors, one of which is free to move and the other one is fixed. The incoming light is confined and mul-tiply reflected by the mirrors. The motion of the mirror changes the length of the cavity and hence the cavity resonance frequency. Shift of the cavity resonance frequency modifies the phase or amplitude of the intracavity field. These modifications can be detected when the light escapes the cavity.

When a photon is reflected by the cavity mirror, momentum is transferred to the mechanical oscillator, which affects the mechanical oscillator by shifting resonance frequency and changing the quality factor. For high input laser powers this back-action effect can disturb the measurement process and must be taken into account. For low input laser shot noise plays an important role which is caused by the random arrival of photons at the detector. Increasing the power of the input laser reduces shot noise but increases the backaction effect on the mirror.

Sensitive detection and actuation of mechanical motion depend upon dy-namical backaction, resulting from the position-dependent feedback of electro-magnetic wave momentum. When the mirror is displaced the intracavity ﬁeld amplitude changes, which in return changes the radiation pressure force expe-rienced by the mirror. This is called dynamical backaction and if the photon life time in the cavity is longer than the period of the mechanical oscillations,

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6 1. Introduction

(a) (b)

Figure 1.2: (a) Schematic picture of the SQUID with suspended beam. Change of magnetic ﬂux induced by the displacement of the beam is detected by measuring the output voltage. (b) Piezoresistive cantilever. Displacement of the cantilever changes the electrical resistance between the two sides. The ﬁgures are reproduced from [36,37].

it can be used to excite and damp oscillations of a mechanical resonator in a controllable manner. This technique is used for self-cooling the mechanical system down to its quantum ground-state.

1.1.2 Nanoelectromechanical-system based realisations

Another class of devices are nanoelectromechanical systems, in which the me-chanical oscillator is coupled to an electronic circuit. Electromeme-chanical sys-tems in general transform energy from mechanical into electrical and vise versa. There are different ways to couple mechanical and electrical degrees of freedom. One way is to use superconducting quantum interference devices (SQUIDs). SQUIDs are very sensitive detectors used to measure extremely weak magnetic fields, based on superconducting loops containing Josephson junctions. The voltage across the loop depends on the amount of magnetic flux that threads the loop. If part of the loop is suspended the area of the loop and thereby the flux through the loop may vary. The flux variation results in modification of the voltage drop across the loop which can be monitored. This scheme is used in Ref. [36] to detect the driven and thermal motion of a 2-MHz buckled-beam resonator (see Fig. 1.2(a)) with femtometre resolution at milliKelvin temperatures.

A diﬀerent way to convert displacement into an electrical signal is to use piezoelectricity, ﬁgure 1.2(b), i.e. materials in which applied stress results in a charge accumulation. In a piezoelectric resonators displacement causing

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1.2. Theoretical concepts 7

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Figure 1.3: Suspended carbon nanotube as the movable island of single electron transistor, reproduced from [38]

additional stress results in a voltage diﬀerence across the resonator. Hence resonators detect their own motion.

Self-detection is also possible when the resonator is a moveable island of a single electron transistor(SET)[38, 39], in which island is a metallic clus-ter which is connected to two electrodes by tunnel junctions. The island is capacitively coupled to a third gate electrode, which is used to change the potential on the island. If the resistance of the junctions is larger than the resistance quantum h/e2 and the total capacitance of the island is so small that associated charging energye2/Ct ot is larger than the thermal energykbT,

then the charge on the island is quantized and electrons will tunnel in and out of the island one by one. Now if the island is moving such a way that the capacitance between the gate and the island is modiﬁed, then the potential on the island changes, thereby modifying the (observable) electric current ﬂowing through the island. This method was used in several experiments and a strong coupling between mechanical motion and the charge on the island has indeed been observed[38,39].

1.2 Theoretical concepts

I

n this section some useful theoretical concepts, used in this thesis, are dis-cussed.

1.2.1 Harmonic Oscillator

Mechanical resonators can be found in various devices in a wide range of sizes. The simplest way to describe the motion of the resonator is to model it as a harmonic oscillator. The theory of the harmonic oscillator and vibrational motion in general is a key to understanding a wide range of physical

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phenom-{ {

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8 1. Introduction F=0 F F x=0 x>0 x< 0

Figure 1.4: Example of a harmonic oscillator: spring–mass system in stretched(top), equi-librium(middle) and compressed(bottom) states.

ena. It has great importance both in classical and quantum mechanics. Below we brieﬂy review both.

Classical Mechanics

The classical harmonic oscillator is a system that experiences a restoring force

⃗Fproportional and opposite to a displacement⃗xfrom its equilibrium position. The best example is a mass mconnected to a spring of negligible mass that is ﬁxed at the other end. For simplicity we can add a constraint that it may only move in one dimension. The importance of this model lies in the fact that equations of a similar form arise when studing diﬀerent problems such as: vibrations of strings or drum heads, a particle moving through any region whose potential has one or more local minima (planetary and satellite motion, the classical description of an electron in orbit around a nucleus, pendulums, etc.), LCR circuits, etc..

Beginning with the Newton’s second law of motion and ignoring dissipative forces such as friction and air resistance, we can derive a second order linear diﬀerential equation:

¨

x+ ω20x= 0, (1.1)

whereω0is natural frequency given by the massmof the object and the spring

constant ksp, ω0=

√

ksp/m. Solving this equation gives us the displacement

of the mass as a function of time:

x(t )= A0sin (ω0t+ ϕ). (1.2)

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displacement and velocity of the system. From the equation of motion it is easy to show that the quantity:

E≡m ˙x 2 2 + mω2 0x 2 2 (1.3)

is constant. This is the energy of the simple harmonic oscillator and it is intuitive that without damping it is conserved. The ﬁrst and the second term correspond to the kinetic and potential energies, respectively. When a spring is stretched or compressed, it stores elastic potential energy, which then is transferred into kinetic energy such that the sum is constant.

A more realistic physical model includes dissipation: the damped harmonic oscillator. A good approximation for most environments is proportionality of the dissipative force to the velocity of the mass directed into the opposite direction. This leads to the diﬀerential equation:

¨

x+ η ˙x + ω2₀x= 0, (1.4) with the damping constant η. This is the only type of dissipative force for which the equation of motion has a known exact solution:

x(t )= A0e− η 2t_{sin (}ω₁_t+ ϕ₀_), (1.5) whereω2 1= ω 2 0− (η/2)

2_{is damped frequency. The amplitude} _A

0and the phase

ϕ0 again depend on the initial conditions. We can now explore the behavior

for diﬀerent parameter values. When the damping factor vanishes the system reduces to the simple harmonic oscillator that oscillates at the natural fre-quency with constant amplitude. When the damping factor is larger than zero the system may or may not oscillate, depending on the relation between the damping constantηand the natural frequencyω0.

• The system is underdamped when ω0>η₂ (case of small damping). It

oscillates with the reduced frequencyω1and the amplitude exponentially

decreases to zero. So if we wait long enough the system will settle into its equilibrium position;

• Critically damped case is when ω0=η2. The system does not oscillate

and returns to equilibrium on the scale of the period of oscillation. Since this yields the fastest possible equilibration, it is a regime of interest for several practical applications;

• The system is overdamped when ω0< η₂. The system returns to its

equilibrium position without oscillation but slower than in the critically damped case.

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10 1. Introduction

Another interesting problem arises when a damped harmonic oscillator is driven by an externally applied time-dependent force: the driven harmonic oscillator. When the driving force is of the form:

F (t )= F0sin(ωdt ), (1.6)

then the equation of motion

¨

x+ η ˙x + ω20x=

F0

msin(ωdt ) (1.7)

has an exact solution:

x(t )= A0e− η 2t_{sin (}ω₁_t+ ϕ₀₎+ A sin(ω_d_t+ φ), (1.8) where A= F0 m√(ω2₀− ω2_d)2+ η2_ω2 d ; _{φ = arctan} ηωd ω2 d− ω 2 0 . (1.9)

This solution has two parts: a transient and a steady state. The former is the solution of the damped harmonic oscillator, with exponentially decaying amplitude, and the prefactorA0and phaseϕ0deﬁned by the initial conditions.

The steady state solution has an amplitude that remains constant no matter what initial conditions the oscillator had and it is oscillating with driving frequency ωd. So the behavior of the system in the long time limit is thus

determined by the driving force.

The amplitude acquired by the oscillator eventually, depends on the damp-ing coeﬃcientηand the ratio between the driving frequencyωd and the

nat-ural frequency ω0 of the oscillator. When the driving frequency is equal to

ωr=

√

ω2

0− η2(the resonance frequency) the amplitude reaches its maximum

valueF0/mηω0and it is said that the system is on resonance. This resonance

eﬀect only occurs when ω0> η, i.e. for suﬃciently underdamped systems.

In summary, we have seen how a second order linear diﬀerential equation, describing the simple harmonic oscillator, can generate a variety of behaviors. For the damped harmonic oscillator there is exponential decay to an equilib-rium position. The determining factor that describes the system is the relation between the natural frequency and the damping factor. In the driven harmonic oscillator we see transience leading to steady state periodicity. The asymptotic behavior of the system depends on the relation between the driving frequency and the natural frequency. At the resonance frequency, even small periodic driving forces can produce large amplitude of the oscillations.

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Quantum Mechanics

The harmonic oscillator plays a key role in quantum physics as well. It is one of the few quantum-mechanical systems for which the (descrete) energy spectrumEn and wave functionsψn(x)are known.

In the following we concentrate again on the one-dimensional case for which the classical Hamiltonian function of a particle with mass moscillating with frequencyω0has the form:

H= p 2 2m+ mω2 0x 2 2 , (1.10)

where p is the momentum of the particle. The corresponding quantum-mechanical Hamiltonian reads

ˆ H= pˆ 2 2m+ mω2 0xˆ2 2 , (1.11)

where in the position representation pˆ= −iħd/dx and xˆ represents multipli-cation by x. These operators satisfy the commutation relation: [x, ˆˆ p]= iħ, whereħis the Planck constant. We now introduce two new operators:

ˆ a = ( mω0 2ħ )1/2 ˆ x+ i ( ₁ 2mω0ħ )1/2 ˆ p; ˆ a† = ( mω0 2ħ )1/2 ˆ x− i ( ₁ 2mω0ħ )1/2 ˆ p, (1.12)

one being the Hermitian conjugate of the other. We can also deﬁne a number operator _Nˆ _{= ˆa}†_a_ˆ_{. The Hamiltonian can be expressed in terms of number}

operator as: ˆ H= ħω0 ( ˆ N+1 2 ) . (1.13)

The eigenstates of_Hˆ _{must also be eigenstates of}_Nˆ_{, so we denote those}

(normal-ized) orthogonal eigenstates by |n〉, such that _Nˆ_{|n〉 = n |n〉} _and _Hˆ_{|n〉 = E}_n_|n〉

whereEn= ħω0

(

n+1₂)is the energy ofn-th eigenstate. These operators satisfy the following commutation relations:

[ ˆ

a, â†]= 1; [N , ˆˆ a†]= â†; [N , ˆˆ a]= − â.

After normalization of eigenstates one can show that the action of aˆ on state |n〉 produces the state |n − 1〉 multiplied by the constant pn and aˆ† acts on |n〉 producing|n + 1〉 multiplied by the constant pn+ 1. For this reason, aˆ is

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12 1. Introduction Y₀HxL Y₁HxL Y₂HxL Y₃HxL Y₄HxL Y₅HxL Y6HxL E0 E1 E2 E3 E4 E5 E6 -₃ -₂ -₁ ₀ ₁ ₂ ₃ 0 1 2 3 4 5 6 7 xxzpm E Ñ Ω0

Figure 1.5: Energy levels and correspondent wave functions of the harmonic oscillator for n=0..6.

called an annihilation operator, andaˆ† a creation operator. The two operators together are called ladder operators.

The displacement is described by a wave functionψ(x)which is the solution of the time-independent Schrödinger equation:

ˆ Hψ = −ħ2 2m ∂2_ψ ∂x2+ 1 2mω 2 0xˆ2ψ = Enψ. (1.14)

The wave functions ψn(x) corresponding to the En eigenenergies can be

found explicitly and are depicted in Fig. 1.5. The square of the absolute value of the wave function| ψn(x)|2gives the probability density of ﬁnding the

resonator in positionx given that it is in state|n〉. In quantum mechanics the lowest possible energy is nonzero, the so called zero-point energy E= 1₂ħω0.

So when the oscillator is in its lowest possible energy state (ground state) it still moves around the potential minimum and the standard deviation of the ground state probability density gives the zero-point motion:

xzpm= 〈0| ˆx2|0〉1/2=

√

ħ

2mω0

. (1.15)

So far we used only eigenstates of the energy operator of the harmonic oscillator, which yields insights into the energy spectrum of the system. But if we want to see the connection to the classical limit the energy eigenstates are not the best choice. In elementary discussions of quantum mechanics, the limit of large quantum numbers is often identiﬁed as the classical limit, leading to the so called correspondence principle. Expectation values of the position and

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the momentum operators in energy eigenstates are time independent, so we can evaluate the expectation values ofxˆ and pˆ at any time we wish and they will have the same value in time. These expectation values are 〈 ˆx〉 = 〈 ˆp〉 = 0 and for bign they remain to be zero. however, expectation values ofxˆ and pˆ

in classical limit are expected to satisfy a classical equation of motion. A coherent state is a speciﬁc quantum state of the quantum harmonic oscillator whose dynamics more closely resembles the oscillating behavior of a classical harmonic oscillator. Mathematically, the coherent state|α〉is deﬁned to be the eigenstate of the annihilation operatoraˆ:

ˆ

a|α〉 = α|α〉, (1.16)

where αis a complex number. It is possible to normalize coherent states as

follows: _∫

d2α

π |α〉〈α| = 1. (1.17)

The overlap between diﬀerent coherent states, | 〈α|β〉 |2_{= e}−|α−β|2

is nonzero forα,β. The set|α〉is therefore overcompleted. Expanding into the energy states:

|α〉 = e−12|α|2 ∞ ∑ n=0 αn p n!|n〉 = e α ˆa†_−α∗_a_ˆ |0〉. (1.18)

In the last part the unitary operator of a displacement is introduced which creates a coherent state from the vacuum.

Time evolution of a coherent state is given by the time evolution operator U (t,t′₎_{= e}−i

ħH (tˆ −t′). If we assume that att′= 0the state was|α(0)〉then after some timet the evolution operator will bring it to the state:

|α(t)〉 = U (t,0)|α(0)〉 = e−ħiH (t )ˆ e−12|α(0)|2 ∞ ∑ n=0 αn₍₀₎ p n! |n〉. (1.19) But the set|n〉 is composed of the eigenvectors of the Hamiltonian_Hˆ_{, so:}

|α(t)〉 = e−12|α(0)|2_e−i2ω0t∑∞ n=0 (_α(0)e−iω0t_a_ˆ†₎n p n! |0〉 = e−12|α(0)|2−i2ω0t+α(0)e−iω0taˆ†_|0〉. _(1.20)

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14 1. Introduction

If we compare this expression to (1.18) it becomes obvious that the ﬁrst and the third terms in the exponent after acting on the ground state will give us a coherent state with time-dependent eigenvaluese−iω0tα(0)_{while the second} term will only contribute as a phase factor. Thus we have:

|α(t)〉 = e−i2ω0t¯¯¯e−iω0tα(0)

⟩

. (1.21)

So the coherent state remains coherent under time evolution. Furthermore,

α(t) = e−iω0tα(0) ⇒ d

d tα(t) = −iω0α(t) (1.22) or in terms of real and imaginary parts ofα:

d

d tℜ(α) = ω0ℑ(α); d

d tℑ(α) = −ω0ℜ(α). (1.23) From (1.12) can be shown that:

ℜ(α) = √ mω0 2ħ 〈α|x |α〉; ℑ(α) = √ 1 2mħω0〈α|p |α〉. (1.24)

Deﬁning the expectation values as〈α|x |α〉 = x(t)and〈α|p |α〉 = p(t), and using the latter equations we get:

d d tx(t ) = √ 2ħ mω0 d d tℜ(α) = √ 2ħ 2mω0ω0ℑ(α) = p(t ) m ; d d tp(t ) = √ 2mħω0 d d tℑ(α) = − √ 2mħω0ω0ℜ(α) = −mω20x(t ), (1.25)

i.e. x(t ) and p(t ) satisfy the classical equations of motion. It is also easy to check that coherent states satisfy the minimum uncertainty relation:

⟨

(∆x)2⟩_α⟨(∆p)2⟩_α=ħ

4, (1.26)

where ⟨(∆x)2⟩_α and ⟨(∆p)2⟩_α are the uncertainties of the position and the momentum in the coherent state |α〉.

Due to these properties the coherent states are very natural for studying the classical limit of quantum mechanics.

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(a) (b)

Figure 1.6: (a) Schematic representation of the single electron transistor. (b) Coulomb diamonds. The transport is blocked inside white diamonds. The light brown region denotes where single electron transport (SET) takes place.

1.2.2 Coulomb Blockade

T

he Coulomb blockade is a phenomenon which arises in nanostructures as a result of electron-electron interaction. In solid state physics electrons in conductors are usually considered to be non-interacting quasi particles, form-ing the so called Fermi liquid. These quasi particles are elementary excitations above the (interacting) ground state that do not interact if their energies are close to Fermi energy, which makes the non-interacting particle model good approximation. However, as the size of the conductors decreases interaction cannot be neglected anymore, which bring us to the so called Coulomb Block-ade regime. The behavior of an isolated conducting cluster is now deﬁned by charge quantization, which gives rise to a charging eﬀect.

A suitable model system to investigate the Coulomb blockade in more detail consists of a confined region (small island) weakly coupled by tunnel junction to two leads (source and drain). A tunnel junction is a quantum mechanical circuit element consisting of two conductors separated by an insulator. It can be characterized by the junction capacitanceC and the junction resistanceR. In contrast to a classical capacitor a tunnel junction is leady, i.e. electrons can tunnel through the insulator. Also the resistance of the tunnel junction fundamentally differs from an Ohmic resistor. An additional gate electrode, capacitively coupled to the island, can be used to control a potential on the confined region.

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{ {

1

16 1. Introduction

If the resistance of the junctions is high enoughii _{such that the charge on}

the island is suﬃciently isolated, the charge must be an integer number of elementary charges Q= ne, n being the number of excess electrons on the island and e the electron charge. Due to the small size of the island the Coulomb interaction between charges on the island becomes important. So before tunneling onto the island, an electron has to overcome the Coulomb repulsion by the electrons on the island. This leads to an energy gap between diﬀerent charge states of the island, which is called charging energy:

Ech=

e2

2Ct ot

, (1.27)

whereCt ot= Cg+CL+CR is the total capacitance of the island.

To analyze electron transport, one should specify energy diﬀerences be-tween ﬁnal and initial states characterizing each tunneling event, as con-tributed by the change of electrostatic energy and the work done to add or extract an electron from the island:

∆E±L,R= 2Ech ( n+1 2− CgVg+CLVL+CRVR e ) ∓ eVL,R. (1.28)

Here+(−) represents addition (extraction) of an electron. The voltage applied to the left (right) electrode is denoted byVL (VR) andVg is the voltage at the

gate electrode. At zero temperature tunneling electrons cannot gain energy from the thermal bath. Electrons can then tunnel through the island only when the corresponding energy diﬀerences are negative, ∆E < 0. In those re-gions of the parameter space (deﬁned by gate and transport voltages), where

∆E > 0, tunneling current is suppressed. These regions are characterized by a ﬁxed number of discrete charges on the island and are called Coulomb dia-monds, ﬁgure 1.6(b). Between two diamonds only one charge state is inside the bias window and the single electron transport can take place. In this regime electrons tunnel from source to drain one by one.

The energy scales that play an important role in the Coulomb blockade regime are: the charging energyEch, measuring the strength of the Coulomb

interaction between charges on the island which depends on the geometry of the island; the thermal energy, kBT, deﬁning the width of the Fermi-Dirac

distribution in the charge reservoirs; the mean level spacing δ, which is the energy spacing between the single electron levels on the island; the intrinsic

ii_{From Heisenbergs uncertainty relation follows that the conductance}_G_{of the junction should}

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1.3. Outline 17

{ {

1

quantum broadening of electron levels,ħΓ, set by ﬁnite lifetimeτ = 1/Γ of an electron on the island.

Depending on these values of the energy scales two diﬀerent regimes can be distinguished where the Coulomb blockade and single-electron charging phenomena are observed. The ﬁrst caseEch≫ kBT≫ δ ≥ ħΓis called Classical

regime of the Coulomb blockade. In this limit the electron energy spectrum can be regarded as continuous and the occupation of electron states on the island follows the Fermi-Dirac distribution. In the classical regime the system depicted on Fig. 1.6(a) is referred to as Single Electron Transistor. When Ech≥ δ ≫ kBT ≥ ħΓ the island is in the quantum regime. In this case the

discreteness of the electron levels in the island cannot be ignored which will also contribute to the addition energy. In this regime above described system is referred to as Quantum dot.

1.3 Outline

This thesis focusses is on the electrical and mechanical characteristics of na-noelectromechanical systems (NEMS) and is organized in the following way: Chapter 2 discusses modulation of the electric current by ac-driving of NEMS. Modiﬁcation of the mechanical quality factor of the oscillating part of NEMS in a single electron transport regime and the mechanism of dissipation is dis-cussed in Chapter 3. Interaction between vibrational modes of double-clamped mechanical resonators due to nonlinearities is studied in Chapters 4 and 5. Chapter 4 is devoted to the classical description of this interaction and Chap-ter 5 addresses the quantum dissipative dynamic of the system.

References

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[4] R. Feynman, There’s Plenty of Room at the Bottom, Engineering and Science 23(5), 22 (1960).

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18 References

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[7] F. Haldane,‘Luttinger liquid theory’ of one-dimensional quantum ﬂuids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas, J. Phys. C: Solid State Phys 14, 2585 (1981).

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[9] K. v. Klitzing, G. Dorda, and M. Pepper,New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45, 494 (1980).

[10] B. Al’tshuler,Fluctuations in the extrinsic conductivity of disordered con-ductors, Phys. Rev. Lett. 41, 648 (1985).

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[12] Y. Aharonov and D. Bohm,Signiﬁcance of Electromagnetic Potentials in the Quantum Theory, Phys. Rev. 115, 485 (1959).

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[14] M. Li, H. X. Tang, and M. L. Roukes, Ultra-sensitive NEMS-based can-tilevers for sensing, scanned probe and very high-frequency applications, Nature Nanotech. 2, 114 (2007).

[15] Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, Zeptogram-Scale Nanomechanical Mass Sensing, Nano Letters 6, 583 (2006),http://pubs.acs.org/doi/pdf/10.1021/nl052134m.

[16] K. L. Ekinci, X. M. H. Huang, and M. L. Roukes, Ultrasensitive nano-electromechanical mass detection, Appl. Phys. Lett 84, 4469 (2004).

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[17] K. L. Ekinci, Y. Y. Yang, and M. L. Roukes,Ultimate Limits to Inertial Mass Sensing Based upon Nanoelectromechanical Systems, J. Appl. Phys

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[19] Y. Cui, Q. Wei, H. Park, and C. M. Lieber, Nanowire Nanosensors for Highly Sensitive and Selective Detection of Biological and Chem-ical Species, Science 293, 1289 (2001), http://www.sciencemag.org/ content/293/5533/1289.full.pdf.

[20] K. Besteman, J.-O. Lee, F. G. M. Wiertz, H. A. Heering, and C. Dekker,Enzyme-Coated Carbon Nanotubes as Single-Molecule Biosen-sors, Nano Letters 3, 727 (2003), http://pubs.acs.org/doi/pdf/10. 1021/nl034139u.

[21] W. Zhao, S. Fan, N. Xiao, D. Liu, Y. Y. Tay, C. Yu, D. Sim, H. H. Hng, Q. Zhang, F. Boey, et al.,Flexible carbon nanotube papers with improved thermoelectric properties, Energy Environ. Sci. 5, 5364 (2012).

[22] K. Besteman, J.-O. Lee, F. G. M. Wiertz, H. A. Heering, and C. Dekker,Enzyme-Coated Carbon Nanotubes as Single-Molecule Biosen-sors, Nano Letters 3, 727 (2003), http://pubs.acs.org/doi/pdf/10. 1021/nl034139u.

[23] H. W. C. Postma, T. Teepen, Z. Yao, M. Grifoni, and C. Dekker,Carbon Nanotube Single-Electron Transistors at Room Temperature, Science 293, 76 (2001).

[24] S. Axelsson, E. E. B. Campbell, L. M. Jonsson, J. Kinaret, S. W. Lee, Y. W. Park, and M. Sveningsson, Theoretical and experimental investi-gations of three-terminal carbon nanotube relays, New Journal of Physics

7, 245 (2005).

[25] A. Subramanian, A. R. Alt, L. Dong, B. E. Kratochvil, C. R. Bolognesi, and B. J. Nelson,Electrostatic Actuation and Electromechanical Switching Behavior of One-Dimensional Nanostructures, ACS Nano 3, 2953 (2009), pMID: 19739601,http://pubs.acs.org/doi/pdf/10.1021/nn900436x. [26] A. B. Kaul, E. W. Wong, L. Epp, and B. D. Hunt, Electromechanical Carbon Nanotube Switches for High-Frequency Applications, Nano Letters

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20 References

[27] N. R. Wilson and J. V. Macpherson, Carbon nanotube tips for atomic force microscopy, Nature Nanotechnology 4, 483 (2009).

[28] J. Kim, J. Park, and C. Han, Use of dielectrophoresis in the fabrication of an atomic force microscope tip with a carbon nanotube: experimental investigation, Nanotechnology 17, 2937 (2006).

[29] A. D. O�Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, et al., Quantum ground state and single-phonon control of a mechanical resonator, Nature

464, 697 (2010).

[30] J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whit-taker, and R. W. Simmonds,Circuit cavity electromechanics in the strong-coupling regime, Nature 471, 204 (2011).

[31] A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, Observation of Quantum Motion of a Nanomechanical Resonator, Phys. Rev. Lett. 108, 033602 (2012).

[32] E. Verhagen, S. Deleglise, S. Weis, A. Schliesser, and T. J. Kippenberg, Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode, Nature 482, 63 (2012).

[33] K. L. Ekinci and M. L. Roukes, Nanoelectromechanical systems, Review of Scientiﬁc Instruments 76 (2005).

[34] T. J. Kippenberg and K. J. Vahala, Cavity Opto-Mechanics, arX-ive:0712.1618 (2007).

[35] F. Marquardt and S. M. Girvin,Optomechanics, Physics 2, 40 (2009). [36] M. Poot, S. Etaki, I. Mahboob, K. Onomitsu, H. Yamaguchi, Y. M.

Blanter, and H. S. J. van der Zant,Tunable Backaction of a DC SQUID on an Integrated Micromechanical Resonator, Phys. Rev. Lett. 105, 207203 (2010).

[37] M. Poot and H. S. J. van der Zant, Mechanical systems in the quantum regime, arXive:1106.2060v2 (2011).

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[39] A. K. Hüttel, G. A. Steele, B. Witkamp, M. Poot, L. P. Kouwenhoven, and H. S. J. van der Zant,Carbon Nanotubes as Ultrahigh Quality Factor Mechanical Resonators, Nano Letters 9, 2547 (2009), pMID: 19492820, http://pubs.acs.org/doi/pdf/10.1021/nl900612h.

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2

Current in ac-driven

nanoelectromechanical

systems

G. Labadze and Ya. M. Blanter

We investigate electric current in a single-electron tunneling device weakly coupled to an ac-driven underdamped harmonic nanomechanical oscillator. In the linear regime, the current can respond to external frequency in a resonant as well as in an anti-resonant fashion. We have found that in the case of the exponential energy dependence of the tunneling rates one can have a strong feedback regime, which results in a modiﬁcation of the current response to external driving.

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24 2. Current in ac-driven nanoelectromechanical systems

2.1 Modeling of NEMS

N

anoelectromechanical systems can be modeled as a single electron tran-sistor (SET) or a quantum dot coupled to a mechanical oscillator. The coupling between the charge of the SET (quantum dot) and mechanical de-grees of freedom originates from the dependence of the charge state of the island on the position of the mechanical oscillator.

The system we describe here is made of two leads (source and drain) con-nected to a movable island via tunnel barriers. The mass and the natural frequency of the resonating island are denoted by M and ω0. The island is

capacitively coupled to the third (gate) electrode. The most obvious speciﬁc realization is a suspended beam (carbon nanotube), see ﬁgure 2.1(a), between two electrodes. Bias voltage Vb= VL−VR can be applied between these

elec-trodes which results in an electrical current ﬂow. The gate voltageVg is used

to change potential on the island. Vibrations of the beam modify the gate ca-pacitance. We assume that the oscillator is additionally driven by an external antenna which creates an ac signal with the amplitude Vd and the frequency

ω. Our inspiration to study this particular design came from the experiments performed on suspended carbon nanotubes[1,2].

Figure 2.1(b) shows the equivalent capacitance circuit: The dot represents the island, which is connected to four voltage sources via the capacitances CL,CR,Cg(x),CD. The electrostatic energy of this system can be written in the

following way: Eel= Ech ( n−q(x) e )2 −CLV 2 L 2 − CRV_R2 2 − Cg(x)Vg2 2 − CDV_D2 2 , (2.1) where Ech= e2/2CΣ(x),CΣ(x)= CL+CR+Cg(x)+CD, and q(x)= CLVL+CRVR+

Cg(x)Vg+CDVD are the charging energy, the total capacitance and the induced

charge on the island by the electrodes respectively. The displacement of the island is denoted byx and in what follows we assume that it is much smaller than the distance between the island and the gate electroded1and both these

distances are smaller than the distance between the island and the antenna d2 (x≪ d1≪ d2). This means that the gate capacitance is much larger than

the capacitance between the island and the antennaCD≪ Cg and it is possible

to neglect the position dependence of CD. Expanding the gate capacitance

close to its equilibrium position (in our case we assume that the equilibrium position corresponds to the zero displacement), up to the ﬁrst order in the displacement results in:

Eel= E0ch ( n−q0 e )2 − Fel(n)x− CLVL2 2 − CRVR2 2 − Cg0Vg2 2 − CDVD2 2 , (2.2)

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2.1. Modeling of NEMS 25 { {

2

Vg VL VR xHtL VD (a) VR VL ~ Vg VD CD Cg CL CR (b)

Figure 2.1: (a) Illustration of the setup: a beam is suspended over two electrodes and can oscillate freely. The beam is capacitively coupled to the gate electrode. A bias voltage

Vb= VL−VR is applied to the electrodes, and the beam is driven by the ac signal applied to

the antenna. (b) Schematic version of setup: the dot in the center corresponds to the beam which is connected to four voltage sources via corresponding capacitances.

where nowE_ch0 = e2/2CΣ(0),CΣ(0)= CL+CR+Cg(0)+CD, andq0= CLVL+CRVR+

Cg(0)Vg+CDVD are the charging energy, the total capacitance and the induced

charge on the island for zero displacement. n represents the charge state of the island, andFel(n)is the force acting on the island from all four electrodes,

including the driving force. Taking the driving force of the formVD= Vdsin(ωt)

gives: Fel(n)= f1(V, n)+ f2(V, n) sin(ωt) + f3(V, n) sin2(ωt), (2.3) with f1(V, n) = [(_C L(VL−Vg)+CR(VR−Vg)−CDVg−en CΣ )2 −V 2 g 2 ] ∂Cg ∂x ; f2(V, n) = 2 ( CL(VL−Vg)+CR(VR−Vg)−CDVg−en ) CDVd C_Σ2 ∂Cg ∂x ; f3(V, n) = C_D2V_d2 C2_Σ ∂Cg ∂x .

The charge transfer process at a given n is characterized by the energy difference between final and initial states, which is the difference in electrostatic energies given by Eq.(2.2) plus (minus) an energy cost associated with addition (extraction) of an electron to (from) the corresponding electrode:

∆En→n+1 L = −2Ech0 ( n+1 2− q0 e ) − Fstx− eVL; ∆En+1→n R = 2Ech0 ( n+1 2− q0 e ) + Fstx+ eVR, (2.4)

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{ {

2

whereFst≡ Fel(n+1)−Fel(n)is a stochastic force acting on the island caused by

adding an extra charge on it. If the electrons tunnel one by one from source to drain electrode, then the island will experience a force in the form of a random telegraph signal.

It is important to specify the regime used in our approach. Concerning the time scales, it is considered that during an oscillating period there are many tunneling events meaning that the typical electron tunneling rate Γ is much higher than the oscillation frequencyω0< Γ. We also consider that the

oscillator is underdamped, which means that time scale set by the mechan-ical damping parameter ω0/Q is largest. Here Q is the quality factor of the

oscillator.

The charging energy of the system is assumed to be much larger than the applied voltage, which implies that only two charge states will be important for the transport. We also consider that the smallest energy scale is set by the temperature of the system, kBT. There are two scales for the energy of

the oscillations: the quantum energy scale EQ= ħω0 and the energy scale to

aﬀect the tunneling ratesEΓ which happens when the workFstx is of the same

order as the charging energy E0_ch. The amplitude AΓ of the oscillations with

the energyE_Γ is A_Γ= √ 2E_Γ M 1 ω0 ,

which for the energy scale gives:

E_Γ=E 0 ch 2 E_ch0 Mω2₀ Fst2 =E 0 ch 2 E_ch0 λħω0 , (2.5)

where we have deﬁned:

λ = F 2 st ħMω3 0 . (2.6)

λ is the coupling parameter that tells how much individual tunneling events displace the oscillator. In other words it is the relative shift of the oscillator energy resulting from a single tunnelling event in respect to the zero point motion energy.

Depending on the value of the coupling constant two diﬀerent regimes can be realized. The strong-coupling regime, when λ ≫ 1, is extensively studied in literature. In this regime if the oscillation frequency ω0 is bigger than

the electrons tunneling rate Γ, the behavior of the system is dominated by the Franck-Condon eﬀect i.e. when the tunneling of an electron onto the island with the simultaneous emission or absorption of several phonon is more

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2.1. Modeling of NEMS 27

{ {

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probable than the elastic tunneling. Phonon-assisted electron tunneling leads to the steps in the current as the function of the applied bias voltage [3–5]. The height of the steps in the current is determined by the coupling constant. In the opposite case, whenΓ ≫ ω0, the Franck-Condon structure is smeared out. In

such situation, one can consider electrons in the quasi-stationary ﬁeld potential provided by the oscillator and employ the Born-Oppenheimer approximation. In the references [6–9] authors have shown that the oscillator may become bistable, and the tunneling electron can cause switching events between the two states of the oscillator. This switching events result in the modiﬁcation of the current itself, which is referred to as the strong mechanical feedback.

It is less intuitive that strong feedback is also possible at weak coupling, λ ≪ 1. In reference [10, 11] authors have studied a similar system without external periodic driving in the low-frequency case,ω ≪ Γ. It was reported that the strong feedback regime is feasible, but the behavior of the current strongly depends on the energy dependence of the tunnel rates. For the rates which depend exponentially on the addition energy (2.4) and are asymmetric four different regimes were found: (i) no oscillations are generated; (ii) oscillations are generated with the fixed finite amplitude; (iii) the oscillator can switch between vibrating and non-vibrating states; (iv) the system is bistable with the two states representing the oscillations with two different amplitudes. In the region where the oscillator is in vibrating state the current and the current noise are strongly modified with respect to the case when no phonons are generated. The high frequency case, ω0≫ Γ, was studied in [12]. In this

regime no bistable regions were identified. Only two regions, with zero and finite amplitude, were found. The finite amplitude region exhibited the strong mechanical feedback.

Here we investigate how strong mechanical feedback inﬂuences the response to external periodic driving in the weak coupling regime. We demonstrate that in the linear regime of the oscillations the response can exhibit jump due to strong mechanical feedback.

In the weak coupling regime λ << 1, one can notice that the two energy scales,EΓ andE0_ch, can be of the same order ifE_ch0 >> ħω0(see 2.5). Therefor

we concentrate on the classical limit of our mechanical oscillator.

The random kicks caused by stochastic force (2.6) result in a net energy transfer to the oscillator,d E /d t≃ F_st2/MΓ. From the energy balance between dissipation and this transfer one can write thatd E /d t= Eω0/Q. So for a typical

energy accumulated in the oscillator we getE≃ QF_st2/MΓω0. The amplitude

of the resulting oscillations, A=

√

2E /Mω2₀ can provide strong feedback on the tunneling of electrons ifEch≃ FstA≃ (Fst2/M )

√

Q/Γω3₀= ħω0λ

√

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{ {

2

we concentrate on the classical limit of the oscillator, Ech≫ ħω0, to observe

strong mechanical feedback the quality factor of the oscillator should satisfy the following restriction: Q≫ Γ/(ω0λ2).

2.2 Master equation

E

lectron transport in a single electron transistor (a quantum dot) is very accurately evaluated based on a master equation. Contrary to SET (quan-tum dot) here not only the number of extra charges in the island is relevant to characterize the electrons in a nanostructure but also the coordinate and the velocity of the oscillator plays an important role. So the probability distribu-tion Pn to be in the charge state n depends now on the coordinate and the

velocity. This probability distributionPn(x, v, t )obeys the Boltzmann equation

[13–15], which reads as follows:

{ ∂ ∂t+ v ∂ ∂x+ ∂ ∂v Ft ot(n) M } Pn=St[P ], (2.7)

where we have deﬁned the total force Ft ot(n) acting on the island with the

charge state n. This total force is the sum of the elastic force, the friction force with the damping coeﬃcientγ0 and the electric forces acting from the

electrodes:

Ft ot(n)= −Mω20x− Mγ0v+ Fel(n).

The right-hand side of the equation (2.7) is the ”collision integral” and de-scribes the change of the probability distribution due to electron tunneling:

St[P ]= (2n − 1)(Γ+(x)Pn− Γ−(x)Pn+1). (2.8) HereΓ±_{= Γ}±

L+Γ±R andΓ±L,R are four tunnel rates, where the subscriptsLandR

denote tunneling through the left or right junction, and the superscripts+and −correspond to the tunneling to and from the island, respectively. Each rate is a function of the corresponding addition energy ∆E±L,R associated with the

addition/removal of an electron to/from the island. As these addition energies depend on the position so will the tunneling rates.

In the considered regime only two charge states (n and n+ 1) contribute to electron transport. For an immovable island the left hand side of the Eq. (2.7) is zero and the stationary probability can be written as [16]:

Pn = Γ − Γt ot ; Pn+1 = Γ + Γt ot . (2.9)

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2.2. Master equation 29

{ {

2

If during consequent electron tunneling events the displacement of the island is small, we can employ the adiabatic approximation by the following trans-formation: Pn = Γ − Γt ot P− δp; Pn+1 = Γ + Γt ot P+ δp, (2.10)

where we have introduced the total probability density to have the island at the positionxand with velocityv,P= Pn+Pn+1 andP≪ δp. Using the adiabatic

approximation for the termδp, i.e. taking into account that in zeroth order it does not depend on x and v variables, gives the master equations at this order: Γ− Γt ot { ∂ ∂t+ v ∂ ∂x+ ∂ ∂v Ft ot(n) M } P− v ∂ ∂x Γ− Γt ot P = Γt otδp; (2.11) Γ+ Γt ot { ∂ ∂t+ v ∂ ∂x+ ∂ ∂v Ft ot(n+ 1) M } P− v ∂ ∂x Γ+ Γt ot P = −Γt otδp, (2.12)

Now if we multiply the ﬁrst equation byΓ+ and the second byΓ−and noting thatFt ot(n+ 1) − Ft ot(n)= Fst the subtraction of two equations gives:

vΓt ot [ ∂ ∂x Γ+ Γt ot ] P+Fst M Γ−Γ+ Γt ot ∂P ∂v = −Γ 2 t otδp.

So the zeroth order term ofδpin the adiabatic expansion is: δp = −Fst M Γ−_Γ+ Γ3 t ot ∂P ∂v − v Γ3 t ot [ Γ−∂Γ+ ∂x − Γ+ ∂Γ− ∂x ] P. (2.13)

In the next order of adiabatic expansion we restore the derivatives of theδp most of which can be eliminated by taking the sum of the two master equations. This way we ﬁnally arrive at the equation for the total probability distribution P: ∂P ∂t + v ∂P ∂x+ F (x) M ∂P ∂v = γ(x) ∂(vP) ∂v + D(x) ∂2_P ∂v2. (2.14)

This equation is known as the Fokker-Planck equation [17]. It is similar to the equation describing a motion of the Brownian particle in an external potential. Two terms in the right-hand side of the equation describe a drift and a diﬀusion, which for the Brownian particle is due to the interaction with the surrounding gas.

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The forceF (x)has the form: F (x) = −Mω2

0x2+ Fel(n)+ Γ +

Γt ot

Fst.

It is useful to note that _ΓΓ+

t ot is the stochastic average excess charge on the island in the charge state n which we denote as n¯. In considered voltage regime 0< ¯n < 1. Using this notation we can rewrite the above expression of the force as:

F (x) = −Mω2 0(x2− ¯n

p

λxQ)+ Fn≈ −Mω20x2+ Fn. (2.15)

Here we have introduced the amplitude of zero-point motion xQ=

√

ħ Mω0. As we are interested in the weak coupling (λ ≪ 1) and the classical motion we can neglect the corresponding term in the force.

The diﬀusion term describes the driving of the oscillator by the stochastic forceFst and has the following form:

D(x)=F 2 st M2 ¯ n(1− ¯n) Γt ot . (2.16)

The drift coeﬃcient is given by:

γ(x) = γ0+ Fst

MΓt ot

∂ ¯n

∂x. (2.17)

If this is positive it describes the energy loss of the oscillator. The first term is due to various friction mechanisms (clamping points of the beam, surrounding gas) and the second term is due to the tunneling events. Thus, the flow of the electric current in the system causes extra dissipation in the mechanical motion. We discuss this in Chapter 3 in detail. For certain energy dependence of the tunneling rates and sufficiently high quality factors the drift coefficient can become negative. In this case it describes the accumulation of energy and causes increase in oscillation amplitude.

2.3 Probability distribution

I

n the single electron transport regime between the charge statesnandn+1 the island is subject to a force which is the sum of two terms: the total electric force Fel(n) and the stochastic force Fst due to stochastic electron

(39)

2.3. Probability distribution 31

{ {

2

subject to such force can be written in the following way: x= x0+δx, wherex0

is due to the total electric force and can be obtained by solving the equation of motion, giving: x0= f2(V, n) sin(ωt) M √ ( ω2_{− ω}2 0 )2 +ω20ω2 Q + f3(V, n) sin(2ωt) 8M √ ( ω2_{− (ω} 0/2)2 )2 +ω20ω2 4Q . (2.18)

Here we have dropped the static displacement because we are interested in modiﬁcation of the electrical properties due to oscillations. The second term δx is a stochastic displacement which originates from the stochastic force. One can also deﬁne corresponding stochastic velocityδv. If now we change the variablesδxand δv of the probability distribution to the variablesx and v, then the equation (2.14) can be transformed as:

∂P ∂t + δv ∂P ∂(δx)− ω 2 0δx ∂P ∂(δv)= = D(x0+δx) ∂ 2_P ∂(δv)2+ v0 ( γ(x0+δx) − γ0 ) ∂P ∂(δv)+ γ(x0+δx)∂(δvP) ∂(δv) .(2.19) This equation can be solved for two limiting cases. The ﬁrst is when x0≫

δx, and the dependence of the diﬀusion and drift coeﬃcients on δx can be disregarded. We will use following parametrization:

δx = √ 2E Mω2 0 sinϕ; δv = √ 2E M cosϕ,

where E is the energy and ϕthe phase corresponding to stochastic displace-ment. Since our interest is the interplay between the tunneling and the energy of the oscillator, we will take an average over the phase. We also disregard the phase dependence of the probability distribution, since it changes very little with the phase. Averaging equation (2.14) over the phase ϕ, we obtain the following: ∂P ∂t = ∂ ∂E [ E ( γ(x0)+ D(x0) ∂ ∂E ) P ] . (2.20)

Utilizing the assumption of the high quality factor one can replaceγ(x0) and

D(x0) by their time averaged values. This gives the possibility to derive the

time averaged distribution function as:

P (E ,ω) = P0exp [ −〈γ(x0)〉t 〈D(x0)〉t E ] . (2.21)

Electromechanics of suspended nanowires

E

LECTROMECHANICS OF SUSPENDED

NANOWIRES

E

LECTROMECHANICS OF SUSPENDED

NANOWIRES

Proefschrift

Giorgi L

ABADZE

Contents

1

Introduction

T

1

1

1

1.1

Experimental realization of mechanical

sys-tems

T

1

1.1.1

Cavity based realisations

1

1.1.2

Nanoelectromechanical-system based realisations

1

1.2

Theoretical concepts

I

1.2.1

Harmonic Oscillator

1

1

1

1

1

1

1

1

1.2.2

Coulomb Blockade

T

1

1

1.3

Outline

References

1

1

1

1

2

Current in ac-driven

nanoelectromechanical

systems

G. Labadze and Ya. M. Blanter

2

2.1

Modeling of NEMS

N

2

2

2

2

2.2

Master equation

E

2

2

2.3

Probability distribution

I

2