This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 143.248.221.120

This content was downloaded on 06/04/2017 at 06:46

Please note that terms and conditions apply.

Transition of a nanomechanical Sharvin oscillator towards the chaotic regime

View the table of contents for this issue, or go to the journal homepage for more

2017 New J. Phys. 19 033033

(http://iopscience.iop.org/1367-2630/19/3/033033)

Home Search Collections Journals About Contact us My IOPscience

You may also be interested in:

Self-excitation in nanoelectromechanical charge shuttles below the field emission regime

F Rüting, A Erbe and C Weiss

Synchronizing a single-electron shuttle to an external drive

Michael J Moeckel, Darren R Southworth, Eva M Weig et al.

Self-sustained oscillations in nanoelectromechanical systems induced by Kondo resonance

Taegeun Song, Mikhail N Kiselev, Konstantin Kikoin et al.

Large parametric amplification in an optomechanical system

F M Buters, H J Eerkens, K Heeck et al.

Non-linear dynamics of semiconductor superlattices

Luis L Bonilla and Holger T Grahn

Topical review: spins and mechanics in diamond

Donghun Lee, Kenneth W Lee, Jeffrey V Cady et al.

Tunable coupled nanomechanical resonators for single-electron transport

Dominik V Scheible, Artur Erbe and Robert H Blick

A capacitive silicon resonator with a movable electrode structure for gap width reduction

Nguyen Van Toan, Masaya Toda, Yusuke Kawai et al.

New J. Phys. 19 (2017) 033033 https://doi.org/10.1088/1367-2630/aa652e

PAPER

Transition of a nanomechanical Sharvin oscillator towards thechaotic regime

JoonHyongCho1,Minah Seo1, Taikjin Lee1, JaeHunKim1, SeongChan Jun2, YoungMin Jhon1,Kang-HunAhn3, RobertHBlick4, HeeChul Park5 andChulki Kim1

1 Sensor SystemResearchCenter, Korea Institute of Science andTechnology, Seoul 02792, Republic of Korea2 Department ofMechanical Engineering, YonseiUniversity, Seoul 03722, Republic of Korea3 Department of Physics, ChungnamNational University, Daejeon 34134, Republic of Korea4 Center forHybridNanostructures, UniversitätHamburg, Falkenried 88,HamburgD-20251, Germany5 Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Republic of Korea

E-mail: hcpark@ibs.re.kr and chulki.kim@kist.re.kr

Keywords:nanoelectromechanical system, impact oscillator, chaos

Supplementarymaterial for this article is available online

AbstractWe realize a nanomechanical impact oscillator driven by a radiofrequency signal. Themechanicalimpact of the oscillator is demonstrated by increasing the amplitude of the external radiofrequencysignals. Electron transport in the system is dramaticallymodifiedwhen the oscillator is strongly drivento undergo forced impacts with electrodes.We exploit this nonlinear kind of electron transport toobserve the current response in the chaotic regime.Ourmodel adopting the Sharvin conductance atthemoment of impact provides a description of the rectified current via the impact oscillator in thelinear regime, revealing a path towards chaos.

Impact oscillators are an important class of discrete dynamical systemswhere an oscillator is driven underintermittent or periodic contacts with limiting constraints [1–5]. Although these oscillators show typically linearresponse, the boundary conditions introduce nonlinearity into the system. Studies on the nonlinear behavior ofimpact oscillators have proliferated on different size scales fromoffshore engineering, where ships collidewithfenders [6], to other practical examples of oscillatory systems such as rattling gears, vibration absorbers, andimpact print hammers [7–9]. Recent progress in nanotechnology provided tools to fabricate systems inwhichmechanical degrees of freedomplay an important role in charge transfer, such as single electron shuttles, wherean oscillatingmetallic island impacts with electrodes [10–12]. However, thorough consideration ofmechanicalimpact on themicro- and nanoscale systems has been hindered so far by the limited control of parameters andmeasurement resolution.

A nanomechanical electron shuttle is an electromechanical coupler which is usually characterized bymechanicallymodulated currents; the shuttle transfers electrons fromone electrode to the other viamechanicaloscillations [10–19]. The electron shuttle consists of a nanomechanical resonator supporting ametallic islandwhich oscillates between two electrodes. This operation principle naturally suppresses co-tunneling events inelectron transport, thus enabling to control the current in the single electron limit even at room temperature[10, 17, 20]. Like for the classical pendulum the driving amplitude and frequency of the nanomechanicaloscillator can be tuned by an external radiofrequency source. It was found thatwhen it is driven by a time-dependent bias voltage the electron shuttle operates as a nanomechanical rectifier at fractional frequencies of thefundamentalmode [21, 22]. This rectification is attributed to a nonlinear response of the shuttle.

The question arising now is whether an electron shuttle experiencingmechanical impacts—traced inelectron transport—against the electrodes, leads to a fundamentally different nonlinear kind of electrontransport. This was discussed already in [11]where single electron shuttles having contacts with source or drainare driven by ultrasonic waves.More recently, it was reported that synchronization of the electron shuttle to anexternal drive could be realized assuming inelastic impact events at itsmaximumdisplacements [12]. Alongthis line of thought, we are investigating the nonlinear current response of a nanomechanical impact

OPEN ACCESS

RECEIVED

10October 2016

REVISED

15 January 2017

ACCEPTED FOR PUBLICATION

7March 2017

PUBLISHED

27March 2017

Original content from thisworkmay be used underthe terms of the CreativeCommonsAttribution 3.0licence.

Any further distribution ofthis workmustmaintainattribution to theauthor(s) and the title ofthework, journal citationandDOI.

© 2017 IOPPublishing Ltd andDeutsche PhysikalischeGesellschaft

oscillator—coined Sharvin oscillator—driven by a radiofrequency signal. Interestingly, the current via thesystem is dramaticallymodulated by themechanical impacts of the oscillator. For this, we suggest a theoreticalmodel which reproduces the characteristic features of the rectified currents in different regimes, revealing a pathtoward chaos.

An electron shuttling system is realized in the formof a torsional resonator designed in such a fashion thatthe center ofmass of the system stays fixed tominimize the dissipation [23]. It is fabricated out of an intrinsicsiliconwafer inwhich a 300 nm thick silicon dioxide layer is thermally formed. An amorphous silicon layer with200 nm thickness was deposited on top of the insulating layer using plasma-enhanced chemical vapordeposition. The resonator is defined by an electron beam lithography andTi/Au layers (10/70 nm) arethermally deposited, providing a conduction path. The patterned gold layer works as an etchmask in reactive ionetching. This leads to form a body of the resonator on the substrate by selectively etching out the amorphoussilicon layer down to the silicon dioxide layer. Then thewhole structure is immersed in buffered oxide etchsolution to remove the sacrificial layer of the silicon dioxide underneath the arms of the resonator. For this, theetch rate for the silicon dioxide has been engineered precisely in order tomake sure that the pivot in theresonator remainswith 200 nm in diameter.

Both oscillating arms have dimensions of 1.5 μm in length and 250 nm inwidth. A shuttle at the tip oscillatesbetween two electrodes (seefigures 1(a) and (b)). A schematic diagramof an equivalent circuit for the electronshuttle is shown in the bottomoffigure 1(b)which is comprised of tunable resistances and capacitances. The gapdistances between the resonator and electrodes are in the range of 25 and 40 nm.All the followingmeasurementswere performed in a probe station equippedwith RF (DC to 40GHz) probes under vacuum ( 10 5 - Torr) atroom temperature. The electromechanical response of the shuttling systemunder external bias voltages isinvestigatedwith themeasurement configuration shown infigure 1(e)where external DC andAC voltagesources can be superimposed via a bias tee. The direct current via the shuttle is amplified in a current

Figure 1. (a)A scanning electronmicroscope image of a torsional resonator. (b)A top view of the resonator. An equivalent circuitdiagram is shown at the bottom. (c) Finite element simulation of the fundamental resonancemode of the resonator. The color coderepresents themagnitude of the displacement. (d)Current–voltage plot in a Fowler-Nordheimmanner. It is well traced by the Fowler-Nordheim equation (a solid line in red). Inset shows electricfield distribution around the electron shuttle calculated by finite elementsimulation [24]. Equipotential lines are indicated by solid lines. (e)Measurement setup. The output current via the electron shuttle isamplified by a current preamplifier.

2

New J. Phys. 19 (2017) 033033 JHCho et al

preamplifier. The threshold voltages in both directional currents are different reflecting the asymmetricconfiguration of the shuttling system in terms of the gap distances towards both electrodes. The electricfielddistribution around the electron shuttle under theDCbias voltage of 1 V is calculatedwhere strong fieldenhancement is apparent due to the proximity of the shuttle and electrodes (see the inset offigure 1(d)) [24].Plotting the data in the Fowler-Nordheimmannermanifests the nonlinear current response at different DCvoltageswhere the linear slope at high voltages is observed using the exponent, k of 1.2 in the Fowler-Nordheimequation (k 2= is the original constant) (see figure 1(d)) [25, 26].

Nowwe explore themechanical response of the electron shuttle in the frequency domain. Finite elementsimulation allows us to estimate the fundamentalmode to be at around f=42.4 MHz (see figure 1(c)) [24]. Thecalculated displacement of the shuttle shows that it is highly possible for the shuttle tomake contacts withelectrodes in resonance considering the formed gap distances. As described in [21], nonzero rectified currentsvia the electron shuttle with asymmetric configuration under AC voltages (zeroDCbias voltage) aremonitored(see themeasurement setup infigure 1(e)). The frequency of the time-dependent voltage is swept in the range of31.3 to 450MHz. The amplitude of the AC voltage varies from1.54 to 3.30 V. The overall current spectra aresummarized infigure 2(a). Aswe increase the amplitude of the applied AC voltage, thewhole structure ofArnol’dtongue is revealedwhere color codes represent rectified current amplitudes in severalmechanicalmodes of theoscillator [16, 22, 27].

The current responses at different AC voltages in the fundamentalmode and amode at the frequency of333MHz (marked by inverted triangles infigure 2(a)) are shown infigures 2(b) and (c). As the AC voltageamplitude increases, the rectified currents arise at resonances of the torsional oscillator. At low amplitudes of theappliedAC voltages, the spectra reflect a linear behavior of the system.Wenote, however, that the system at the

Figure 2. (a)Colorscale plot of output currents via the torsional resonator in the frequency domain at different AC voltages, revealingArnol’d tongue [22].Markers (inverse triangles in black) indicate the frequencies at which current–voltage traces are plotted in (b) and(c). (b)Current spectrum in the fundamentalmodewithAC voltages ranging from1.8 to 3.02 V. A single peak arises as the voltageincreases. (c)Current spectrum at around 333 MHzwithAC voltages ranging from1.8 to 3.02 V. Twomechanicalmodes interfereeach other, showing anti-symmetric spectrum in the output current. (see supporting information).

3

New J. Phys. 19 (2017) 033033 JHCho et al

frequency of 333MHzhas undershoots and zero crossings at higher voltages (see figure 2(c)). This becomesmore pronounced as theAC voltage increases, starting out at the voltage of about 2.2 V. This stems from theinterference between two adjacentmechanicalmodes since the coupling between twomechanicalmodesincreases at high voltages (see supplementary information).

Our key result is that the obtained current spectrum isdistorted and the current via the torsional resonator inthehigh voltages appear scattered.Current traces for variousmechanicalmodes are presented infigures 3(a)–(d).Although the threshold voltages for the exponential increase and the following scattered currents are differentdepending on the chosenmechanicalmodes, they start to appear in the range of 2.3 to 2.7 V. The inset infigure 3(a)shows amagnified viewof the transition point according to the appliedACvoltage. It is apparent that increasingtheACvoltage across the resonator brings the system into a nonlinear regime.Wenote that both polarities of thescattered output currents exist at high voltages reflecting thebroken symmetry in the activatedmechanicalmode[16, 21, 22]. This observation,whichwebackupwith theoretical consideration to follow, implies that the electronshuttle having intermittent or periodic impactswith electrodes experiences transitions towards a chaotic regime inthe current response.

We suggest a theoreticalmodel to obtain a better understanding of the obtained results where ananoelectromechanical shuttle is assumed to be a damped harmonic oscillatorwith the fundamentalmode atthe frequency, f0. In themodel we are considering a situationwhere the parametric resonance under time-dependent driving leads tomechanical impacts towards electrodes at high voltages. Essential for thismodel arethe additional external driving forces, F x t t;e ( ( ) ) and F x tb ( ( )). The equation ofmotion between the electrodesreads

mx t m x t m x t F x t t F x t¨ ; , 1e b02g w+ + = +( ) ˙ ( ) ( ) ( ( ) ) ( ( )) ( )

wherem is the effectivemass of the oscillator, γ is a dynamic damping parameter, and 0w is the eigenfrequency ofthe oscillator. The system is driven by an electrostatic force, F x t t Q x t t V t L; ;e = -( ( ) ) ( ( ) ) ( ) , whereQ x t t CV t x t, tanh x a= +( ( ) ) ( ) ( ( ) ), andV t V tsinAC w=( ) ( ) (C is the junction capacitance, L is the gap

Figure 3.Output current versus AC voltage at different driving frequencies of (a) 41.7, (b) 113.1, (c) 333.1, and (d) 406 MHz. The insetin (a) a shows amagnified view of the low voltage regime at 41.7 MHz. The output currents start to increase continuously at the lowvoltage regime and transit to nonlinear current responses. At high voltages the current traces are highly distorted and scattered.Wenote that both polarities of the output currents in (a)–(d) exist at high voltages reflecting the degree of broken symmetry in theactivatedmechanicalmode.

4

New J. Phys. 19 (2017) 033033 JHCho et al

distance between the electrodes, ξ is the characteristic tunneling length, and R R1 2ln r l0 0a = ( ) represents thegeometric asymmetry). Two electrodes are located at positions x L

2= and restrain its oscillation such that

elastic impacts occur. The elastic bouncing force is described by F x t h x tb b3 2g= -( ( )) ( ( )) , with

bE R2

3 1 22g =s-

(R is the radius of the shuttle,σ is the Poissonʼs ratio, andE is the Youngʼsmodulus.), and thedeformation distance, h x t x x t x x t= Q( ( )) ( ( )) ( ( ( ))) (x x t x t L 2= -( ( )) ∣ ( )∣ , and zQ( ) is theHeavisidestep function.) [28].

The time-averaged rectified current is calculated as

IV

R

x tt I x t t t

2sech sin ; d , 2cDC

AC

0ò x

a w= + +⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

( ) ( ) ( ( ) ) ( )

with R e2 202p= as resistance unit, and the contact current, I x t;c ( ). Assuming the Sharvinmodel for the

contact conductancewith the detail in supplementarymaterials, the contact current is described as

I x t tV

RD x t t;

2sin , 3c

jj

AC

0å w=( ( ) ) ( ( )) ( ) ( )

with D x tjG x t e L

G x t R e

1 sech 2

2 1

jL

jx t

1

22

0=

x a- -

+

x

x( ( ))( ( ))( ( ))

( ( ( )) )( ) , and the Sharvin conductance, G x tj R

R h x t h x t4

F

2

02= p

l-( ( )) ( ( ( )) ) ( ( ))

( Fl is the Fermiwave length.) [29].When the amplitude of the oscillation is small enough, x 1x ∣ ∣ ,equation (1) can be reduced into the dampedMathieu equation [30]. The principal instability regime is thenobtained by solving the equationwith two variable expansionmethod.

The calculated currents based on the suggestedmodel as above are plotted in the plane of frequencues andAC voltages infigure 4(a).Wefind a principal instability regime,V V V - +, where

Vm L

C

2 4

3

2

33 . 40

22

02 2

02 2 2w

w lw w w w g= - - -( ) ( ) ( ) ( )

The solid line infigure 4(a) indicates the analytic results of critical voltage as a function of external frequencyfrom equation (4). In the principal instability regime, so-calledArnol’d tongue, the parametric amplificationcauses the instability of the shuttling system and themotion ismode-locked in the dressed resonance frequency,

CV

mL0 02

20

2

wW = +l

. The environmental damping shifts the critical transition point of the bifurcationwith

modified edges in theArnol’d tongue. TheArnol’d tongue is divided into three regions: a continuously varyingcurrent region, a scattered current region, and a region having both features (see figures 4(a) and (b)). Thecurrent trace at 1.050w w = is shown infigure 4(b). Different states along the appliedAC voltages are apparentin the current–voltage curvewhere the transition points correspond to the position of dots in figure 4(a). Theyare a conventional shuttling state (US1), a shuttling statewith periodic impacts where the oscillator issynchronizedwith the driving source (US2), and a chaotic shuttling state with irregular impacts (US3).Corresponding states of the electron shuttle are schematically described infigures 4(c)–(e). The current tracebegins with amonotonic increase in the low voltagesmainly due to the geometric asymmetry (see amagnifiedview of the low voltage regime in the inset offigure 4(b)), while it is known that the electron shuttle in asymmetric probing configuration gives rise to zeroDC current under external AC voltages [22]. The shuttle inthis state (US1) shows a periodic oscillationwithout collisions with electrodes. The electrons transfer only whenthe shuttle is displaced sufficiently for electrons to tunnel into electrodes in proximity. In the following state(indicated asUS2 infigure 4(b)) the oscillation of the shuttle is synchronizedwith the external AC voltage atinteger-multiples of the fundamentalmode frequency and has regular impacts with the electrodes. Here, thecurrent increases exponentially unlike the conventional shuttling state. Finally, scattered currents appear at thehigher voltages (indicated asUS3 infigure 4(b)).We identify the theoretically encountered states of the electronshuttle bymatching the calculated current with the experimental one. The critical voltage and current levels aswell as the transition toward the scattered current are in agreement with the experiment despite of the difficultyin quantifying the environmental parameters of the structure at the nanoscale (i.e. parameters given in thecaption offigure 4(b). Hence, ourmodel adopting the Sharvin conductance at themoment of impact provides agood description of the rectified current via the Sharvin oscillator.

The Lyapunov exponents are calculated alongwith the AC voltages as shown infigure 4(f).We note that theLyapunov exponents become positive in the scattered current state implying that long-termprediction becomesimpossible in this regimewhere small uncertainties are amplified fast. The calculated displacement shows highlyirregular impacts with both electrodes (see figure 4(g)).More insight can be gained from aPoincaré section. ThePoincaré section in this regime is partiallyfilled reflecting the complexity of the system induced by those impactevents (see figure 4(h)) [3, 4]. The successive points are found to hop chaotically over the attractor, exhibitingsensitive dependence on the initial conditions as typically expected in a chaotic state. Since the duration of thecontact is sufficiently short compared to the period of the oscillation, impacts can be assumed to bemomentary

5

New J. Phys. 19 (2017) 033033 JHCho et al

and periodic events.With this, wefind similarities between the observed transtion in our system and the one inthe classical kicked oscillatormodel [31].

In conclusion, our experiment demonstrates how impacts affect the current via a nanomechanical oscillator.We fabricated a torsional resonatorwhich oscillates between two electrodes. Under the application of a time-dependent bias voltage a nonzero rectified current is observed at resonance frequencies. The presented systemstands out by thefinding that the electron shuttling under impacts provides with highly nonlinear current. Atheoreticalmodel for the system assumingmechanical impact and Sharvin conductance is suggested and itreproduces well the experimentally obtained currents via the shuttle. Furthermore, the suggestedmodel predictsthe evolution of the current via the torsional resonator, revealing the possible transition towards chaos. Thesefindings open new perspectives to investigate unexplored regimes of nanomechanical electron transport.

Acknowledgments

Thisworkwas partially supported by theKIST institutional program (Project No. 2E27270), the IBS program(Project No. IBS-R024-D1), and Basic Science Research Program through theNational Research Foundation ofKorea (NRF) funded by theMinistry of Education (2013R1A1A2A10013147). RHB likes to thank theCenter forUltrafast Imaging for support. HCP thanks theKIASCenter for AdvancedComputation for providingcomputing resources.

Figure 4. (a)Phase diagram indicating the instability regimewith a parameter, 0.025 0g w= . The amplitudes of currents arerepresented by different colors. The analytic solutions of equation (4) is shown as a solid line. The area inwhite represents a stableregime (S)where no currentflows. The oscillation starts at the boundary betweenwhite (S) and gray (US1). (b)Calculated currents viaan electron shuttle at different AC voltages along the dotted line infigure 4(a)where I 7.75 10 A0

7= ´ - . The adopted parametervalues are E f f x R0.44, 79 GPa, 0.1 , 2 100 MHz, 0.025 2 , , 5F 0 0 0s l x p g p x x= = = = = ´ = = , and 5 nmx = . Three char-acteristic regimes (US1,US2, andUS3) are identified along the increase of the AC voltage. The transition points between states areindicated by circles in black in figure 4(a). The inset shows amagnified view of the low voltage regime. (c)–(e) Schematics of differentstates of the impact oscillator; (c) conventional shuttling, (d) shuttlingwith periodic impacts, and (e) chaoticmotion of the shuttlewith irregular impacts. (f)Dependence of the Lyapunov exponent on the appliedAC voltage. (g)Displacement of the electron shuttlein a chaoticmotionwhere possible impact events are indicated by arrows in red. The inset shows the change in the slope of x(t) at theimpact point. (h)Poincaré section in the chaotic state of the electron shuttle.

6

New J. Phys. 19 (2017) 033033 JHCho et al

References

[1] Shaw SWandHolmes P 1983Phys. Rev. Lett. 51 623–6[2] MooreDB and Shaw SW1990 Int. J. Non-LinearMechanics 25 1–16[3] Peterka F andVacik J 1992 J. SoundVib. 154 95–115[4] BuddC J and Lee AG1996Proc. R. Soc.A 452 2719[5] DyskinAV, Pasternak E and Pelinovsky E 2012 J. SoundVib. 331 2856–73[6] LeanGH1971Trans. R. Inst. Nav. Archit. 113 387–99[7] Karagiannis K and Pfeiffer F 1991NonlinearDyn. 2 367–87[8] Sharif-BakhtiarM and Shaw SW1988 J. SoundVib. 126 221–35[9] Tung PC and Shaw SW1988 J. Vib. Acoust. Stress Reliab. 110 193–200[10] Gorelik LY, IsacssonA, VoinovaMV,KasemoB, Shekhter R I and JonsonM1998Phys. Rev. Lett. 80 4526–9[11] KoenigDR,Weig EMandKotthaus J P 2008Nat. Nanotechnol. 3 482–5[12] MoeckelM J, SouthworthDR,Weig EMandMarquardt F 2014New J. Phys. 16 043009[13] ErbeA,Weiss C, ZwergerW andBlick RH2001Phys. Rev. Lett. 87 096106[14] ScheibleDV, Erbe A andBlick RH2002Appl. Phys. Lett. 81 1884[15] ScheibleDV andBlick RH2004Appl. Phys. Lett. 84 4632–4[16] KimC, Park J and Blick RH2010Phys. Rev. Lett. 105 067204[17] KimC, PradaMandBlick RH2012ACSNano 6 651–5[18] KimC, PradaM, PlateroG andBlick RH2013Phys. Rev. Lett. 111 197202[19] KimC, PradaM,QinH,KimH-S andBlick RH2015Appl. Phys. Lett. 106 061909[20] KimC,KimH-S, PradaMandBlick RH2014Nanoscale 6 8571–4[21] Pistolesi F and Fazio R 2005Phys. Rev. Lett. 94 036806[22] AhnK-H, ParkHC,Wiersig J andHong J 2006Phys. Rev. Lett. 97 216804[23] ScheibleDV,Weiss C andBlick RH2004 J. Appl. Phys. 96 1757–9[24] COMSOLMultiphysics, ver. 5.1, Burlington,MA,USA, 2015[25] Forbes RG 2008Appl. Phys. Lett. 92 193105[26] KimHS,QinH,WestphallM S, Smith LMandBlick RH2007Nanotechnology 18 065201[27] Arnol’dV I 1983Russ.Math. Surv. 38 215–33[28] Landau LD and Lifshitz EM1986Theory of Elasticity vol 7 3rd edn (Oxford: Elsevier)[29] Sharvin YV1965 J. Exp. Theor. Phys. 48 984–5[30] Butikov E I 2004Eur. J. Phys. 25 535[31] Tel T andGruizM2006Chaotic Dynamics: An Introduction Based onClassicalMechanics (NewYork: CambridgeUniversity Press)

7

New J. Phys. 19 (2017) 033033 JHCho et al