We present here a simple analytical model for self-oscillations in
nano-electro-mechanical systems. We show that a field emission self-oscillator
can be described by a lumped electrical circuit and that this approach is
generalizable to other electromechanical oscillator devices. The analytical
model is supported by dynamical simulations where the electrostatic parameters
are obtained by finite element computations.Comment: accepted in AP

A. Ayari

A. Lazarus

E. de Langre

Grogg D.

P. Manneville

P. Poncharal

P. Vincent

S. Perisanu

S. T. Purcell

T. Barois

English

arXiv

Crossref

Simple modeling of self-oscillations in nanoelectromechanical systems

International audienceWe present here a simple analytical model for self-oscillations in nanoelectromechanical systems. We show that a field emission self-oscillator can be described by a lumped electrical circuit and that this approach is generalizable to other electromechanical oscillator devices. The analytical model is supported by dynamical simulations where the electrostatic parameters are obtained by finite element computations. © 2010 American Institute of Physics

Lazarus, Arnaud

Barois, T.

Perisanu, S.

Poncharal, P.

Manneville, Paul

de Langre, Emmanuel

Purcell, S.T.

Vincent, P.

Ayari, Anthony

HAL-Polytechnique

Simple modeling of self-oscillations innanoelectromechanical systemsA. Lazarus, T. Barois, S. Perisanu, P. Poncharal, Paul Manneville, EmmanuelDe Langre, S.T. Purcell, P. Vincent, A. AyariTo cite this version:A. Lazarus, T. Barois, S. Perisanu, P. Poncharal, Paul Manneville, et al.. Simple modeling ofself-oscillations in nanoelectromechanical systems. Applied Physics Letters, American Instituteof Physics, 2010, 96 (19), pp.193114. <10.1063/1.3396191>. <hal-01021117>HAL Id: hal-01021117https://hal-polytechnique.archives-ouvertes.fr/hal-01021117Submitted on 15 Jul 2014HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Simple modeling of self-oscillations in nanoelectromechanical systemsA. Lazarus, T. Barois, S. Perisanu, P. Poncharal, P. Manneville, E. de Langre, S. T. Purcell, P. Vincent, andA. Ayari  Citation: Applied Physics Letters 96, 193114 (2010); doi: 10.1063/1.3396191 View online: http://dx.doi.org/10.1063/1.3396191 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/96/19?ver=pdfcov Published by the AIP Publishing  Articles you may be interested in In-situ comprehensive calibration of a tri-port nano-electro-mechanical device Rev. Sci. 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Manneville,1 E. de Langre,1S. T. Purcell,2 P. Vincent,2 and A. Ayari2,a1Laboratoire d’Hydrodynamique, École Polytechnique, 91128 Palaiseau, France2Laboratoire de Physique de la Matière Condensée et Nanostructures, Université Lyon 1, CNRS,UMR 5586, Domaine Scientifique de la Doua, F-69622 Villeurbanne Cedex, FranceReceived 19 March 2010; accepted 25 March 2010; published online 14 May 2010We present here a simple analytical model for self-oscillations in nanoelectromechanical systems.We show that a field emission self-oscillator can be described by a lumped electrical circuit and thatthis approach is generalizable to other electromechanical oscillator devices. The analytical model issupported by dynamical simulations where the electrostatic parameters are obtained by finiteelement computations. © 2010 American Institute of Physics. doi:10.1063/1.3396191Nanoelectromechanical systems NEMS1 are under ex-tensive research owing to their potential for radio frequencycommunication and highly sensitive sensors. This research,before becoming applicable, will have to cope with severalmajor issues such as crosstalk. Since the work of Ref. 2, anintriguing class of NEMS has been experimentally demon-strated that could circumvent this drawback by nanoactivefeedback. In contrast to quartz-oscillator like architecture,3there is no need for macroscopic external active circuit sincethe nanodevice itself is placed in a self-oscillating regime.This concept was first theoretically proposed for NEMS byGorelik et al.4 in the specific case of the charge shuttle andis now observed in a large variety of experimentalconfigurations.2,5–9 Although the work of Ref. 2 reachesqualitative agreement between experiment and modeling ofthe self-oscillation phenomenon, it lacks simple argumentsabout the origin of the instability. Here, we derive a simplelinearized model and an equivalent purely electrical circuitthat helps one getting further insight on the way to designand scale down such an oscillator. This model is then vali-dated by dynamical and finite element simulations. The ideaexposed in this article, with minor adaptations, could be use-ful for other experimental geometries.In a typical experiment, a nanowire NW or nanotubewith resistance RNW is attached to a tungsten tip in front ofan anode connected to the ground Fig. 1a. The tip is at anegative dc voltage −Vdc from the ground; electrons areemitted from the apex of the nanowire by field emission andcollected by the anode. The NW starts to oscillate spontane-ously in the transverse direction when Vdc is larger than somevoltage threshold. This system can be modeled by twocoupled differential equations see Eqs. 1 and 2 in Ref.2: first, a mechanical equation that can be linearized as fol-lows:x¨ +0Q x˙ + 02x = HU¯ U , 1where x is the transverse displacement of the apex of the NWcompared to the equilibrium position taken positive whenthe NW approaches the anode, 20 the resonance fre-quency of the mechanical oscillator, Q the quality factor, andH a positive parameter characterizing the actuation strengthby electrostatic forces between the wire and the anode. Theseparameters are supposed to be relatively constant in therange of interest. U¯ is the dc voltage between the NW andthe anode and U the ac voltage. U¯ is not equal to Vdc as aresult of the voltage drop through the nanowire. The linear-ized force is the product of U and U¯ because the electrostaticforce is proportional to the square of the total voltage. Sec-ond, the linearized electrical equation reads the following: IFNU+1RNWU + CU˙ = − IFNxx − CU¯ x˙ , 2where C is the capacitance between the NW and the anode,C its derivative with respect to position, and IFNU+U¯ ;xthe field emission current described by the Fowler–NordheimFN equation IFN=AU+U¯ 22 exp−B / U+U¯ . The xdependence of IFN comes from the field enhancement factor.An important point to notice is that the field emissioncharacteristics depends on two inputs, the apex voltage andits position, in the same way as a transistor or a vacuum tube,but the role of the gate or grid is played by the spatial degreeof freedom x. A simple equivalent electrical circuit is shownin Fig. 1b. The electromechanical resonator is representedaElectronic mail: anthony.ayari@lpmcn.univ-lyon1.fr.CRmLm IFNCmRNWVDCU+UimotIACW tip xe-NWAnode(a)(b)FIG. 1. Color online a Schematic of the experimental configuration andb schematic of the equivalent purely electrical circuit of the self-oscillationof the nanoelectromechanical system of Ref. 2.APPLIED PHYSICS LETTERS 96, 193114 20100003-6951/2010/9619/193114/3/$30.00 © 2010 American Institute of Physics96, 193114-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.104.29.1 On: Fri, 11 Jul 2014 12:52:46by a series resonant circuit in parallel with the capacitor C ofEq. 2. In this well-known analogy, the motional currentthrough the resonant circuit is imot=CU¯ x˙ and the passivecomponents are the motional inductance Lm=1 / HU¯ 2C,the motional resistance Rm=0 / QHU¯ 2C and the motionalcapacitance Cm=HU¯ 2C /02. The voltage across the mo-tional capacitance is proportional to x and can be used as thegate voltage of an equivalent transistor delivering the samefield emission current for a given x and U+U¯ . The transcon-ductance of such transistor is IFN /xHU¯ /02. It brings thegain necessary to sustain the self-oscillation regime and actsas a feedback loop.The main parameter of the self-oscillating circuit is thedriving dc voltage above which the system spontaneouslygenerates the ac signal. In the following, we derive a simpleanalytical formula giving the self-oscillation condition. If thenanowire resistance RNW is smaller than the field emissionresistance IFN /U−1, to first order the voltage at the apexU¯ is Vdc and there is no self-oscillation. We consider theopposite case RNW IFN /U−1 because it gives a simplerformula the general case can be calculated straightforwardlyby the same method. However, when the nanowire resis-tance gets larger more power is dissipated in heating insteadof sustaining the oscillation, so that it might seem optimal tokeep RNW larger than the field emission resistance by lessthan an order of magnitude. A single differential equation ofthe full electromechanical system can be obtained by com-bining Eqs. 1 and 2, as follows:x + x¨1 + 0Q  + x˙0Q + HU¯ 2 ln Cx+ 02+ x02 + HU¯ 2 ln x  = 0, 3where =CIFN /U−1 is the discharge time constant of theelectrical circuit. According to the Routh–Hurwitz criterionthis dynamical system is stable when the following:HU¯ 2  ln x− ln Cx1 + 0Q −0Q 1 + 02 + 0Q   0. 4From this inequality, since C and  increase with x, only thevariation in  with x favors the self-oscillation regime andwe can distinguish between two categories of terms that pre-vent from reaching it: i the variation in the capacitancewith x and ii the relative value of  and 0−1. The latter canbe minimized for 0	1 as long as Q1 our nanowireresonators10 routinely reach Q105. In these conditions, thegeometry of the device should be such that  ln /C /x0 to have a chance to observe self-oscillations. Finally thethreshold dc voltage at the apex for self-oscillation is thefollowing:U¯ so =0QH  ln/C/x , 5and the threshold dc voltage of the power supply is Vsodc=U¯ so+RNWIFNU¯ so ,.In order to check the different hypotheses made, we per-formed numerical simulations and determined the electro-static force, capacitance and field enhancement factor byfinite element methods FEM. The sample is a straight10 m long nanowire of radius 100 nm attached to a metal-lic conical tip in front of a metallic plate perpendicular to theaxis of the tip. The nanowire is initially tilted by 20° com-pared to the cone axis. The sole degree of freedom of thenanowire is this angle that can decrease due the attractiveelectrostatic force between the wire and the metallic plate.The distance between the tip end and the plate is 60 m.The mechanical restoring force is taken from the calculatedrigidity of a clamped free beam with a Young modulus of400 GPa and density of 3200 kg /m3, Q=104 and RNW=1010 	. Further details about the simulations and a morerefined mechanical model can be found in Ref. 11. We firstsimulated the spatial variation in C and  and verified that ln /C /x0 for a wide range of angles around 20°, andestablished that H is changing by less than 15%. The dimen-sionless differential equations were then rewritten, their ei-genvalues computed, and the sign of their real part  scruti-nized. The real part defines the growth rate of the mode andthe solution, which is proportional to expt, decay to zerowhen it is negative, so that the system is stable. On the con-trary, 0 makes the system unstable and leads it into astable self-oscillating regime thanks to nonlinear saturatingterms. The oscillation amplitude gets larger as  increases.Finally, we determined stability maps giving the parameterregions where  is positive and self-oscillations possible.Figure 2 represents the stability map of the system fordifferent applied dc voltages v=Vdc /Vref and different dimen-sionless intrinsic frequencies r=0. Vref=400 V is the volt-age above which RNW stops being negligible when comparedto the field emission resistance. One can point out that ithere is no self-oscillation for v1, ii self-oscillations areeasier at higher v the growth rate is larger and the instabilityregion wider. This validates the statement that for optimalself-oscillations RNW needs to be bigger than the field emis-sion resistance the field emission current increases exponen-tially with v, so that the field emission resistance is smallerfor higher v. This figure also clearly demonstrates that self-oscillations are obtained at easiest for r	1.We also calculated the stability map for various qualityfactors. Equation 5 that determines the boundary betweenthe stable region and the self-oscillation region is in rela-tively good agreement with the results of numerical simula-vr1 1.5 2 2.510−110010110251015x 10−5λFIG. 2. Color online Stability map of a nanowire during field emission forQ=10 000 and different normalized voltages v and dimensionless intrinsicfrequencies r.193114-2 Lazarus et al. Appl. Phys. Lett. 96, 193114 2010 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.104.29.1 On: Fri, 11 Jul 2014 12:52:46tions for high voltage, i.e., when IFN /U1 /RNW. Thisconfirms the validity of the above analytical derivation.Equation 5 shows that, as for any other NEMS device,keeping good performance in this case by maintaining theoperating voltage low at the nanoscale and high frequencyrequires an improvement of the capacitive coupling and thequality factor. Finally a simple scaling calculation shows thatr decreases like the inverse of the apex-anode distance.Downscaling thus helps one to reach the regime where r	1. If this term become too small, or if the resistance of thenanowire or nanotube saturates in the ballistic regime, thedevice can still be operated with the help of an additionalconstant resistance between the dc power supply and RNW.In conclusion, using an electrical equivalent circuit, weshowed that the origin of self-oscillations in field emissionNEMS can be understood in terms of motional capacitanceand spatial variation in the field emission current in a feed-back loop. An equation was derived to determine the thresh-old voltage for self-oscillation and its output confirmed bynumerical and FEM simulations Fig. 2 and 3. We expectthat our simple model will demystify the mechanism respon-sible for self-oscillation in field emission NEMS, as it ap-pears that it can be understood with simple classical electri-cal passive components and one transistor. It appears thenthat geometries like the one of Ref. 6 where the self-oscillation mechanism is not yet clearly identified are indeedvery similar to ours and may be understood within the sameframework. This work opens up perspectives for the controland fabrication of low power nano-oscillators for time baseand ac generators applications.This work was supported by French National ResearchAgency NEXTNEMS under Project No. ANR-07-NANO-008-01 and AUTONOME under Project No. ANR-07-JCJC-0145-01 and Région Rhône-Alpes CIBLE program. The au-thors acknowledge the “plateforme nanofils et nanotubeslyonnaise.”1H. G. Craighead, Science 290, 1532 2000.2A. Ayari, P. Vincent, S. Perisanu, M. Choueib, M. Gouttenoire, V. Bech-elany, D. Cornu, and S. Purcell, Nano Lett. 7, 2252 2007.3E. Colinet, L. Duraffourg, S. Labarthe, P. Andreucci, S. Hentz, and P.Robert, J. Appl. Phys. 105, 124908 2009.4L. Y. Gorelik, A. Isacsson, M. V. Voinova, B. Kasemo, R. I. Shekhter, andM. Jonson, Phys. Rev. Lett. 80, 4526 1998.5H. Kim, H. Qin, and R. H. Blick, New J. Phys. 12, 033008 2010.6D. Grogg, S. Ayoz, and A. Ionescu, Proceedings of the International Elec-tron Devices Meeting IEDM, 2009 unpublished, p. 793.7V. I. Kleshch, A. N. Obraztsov, and E. D. Obraztsova, JETP Lett. 90, 4642009.8G. A. Steele, A. K. Huttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P.Kouwenhoven, and H. S. J. van der Zant, Science 325, 1103 2009.9K. L. Phan, P. G. Steeneken, M. J. Goossens, G. E. J. Koops, G. J. A. M.Verheijden, and J. T. M. van Beek, arXiv:0904.3748v1 unpublished.10S. Perisanu, P. Vincent, A. Ayari, M. Choueib, S. Purcell, M. Bechelany,and D. Cornu, Appl. Phys. Lett. 90, 043113 2007.11A. Lazarus, E. de Langre, P. Manneville, P. Vincent, S. Perisanu, A. Ayari,and S. Purcell, “Statics and dynamics of a nanowire in field emission,”Int.J. Mech. Sci. to be published.1 1.5 2 2.5012345x 104vQ0.511.522.53x 10−4λFIG. 3. Color online Stability map of a nanowire during field emission fora dimensionless frequency r=5 and different normalized voltages v and Q.The solid line represents the self-oscillation threshold determined usingEq. 5.193114-3 Lazarus et al. Appl. Phys. Lett. 96, 193114 2010 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:129.104.29.1 On: Fri, 11 Jul 2014 12:52:46

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