# Simple modeling of self-oscillation in Nano-electro-mechanical systems

**ABSTRACT** 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 APL

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arXiv:1004.1062v1 [cond-mat.mes-hall] 7 Apr 2010

Simple modeling of self-oscillation in NEMS

A. Lazarus, P. Manneville, and E. de Langre

Laboratoire d’Hydrodynamique,´Ecole Polytechnique, 91128 Palaiseau, France

T. Barois, S. Perisanu, P. Poncharal, S. T. Purcell, P. Vincent, and A. Ayari∗

Laboratoire de Physique de la Mati` ere Condens´ ee

et Nanostructures Universit´ e Lyon 1; CNRS,

UMR 5586 Domaine Scientifique de la Doua F-69622 Villeurbanne cedex, France

(Dated: April 8, 2010)

Abstract

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.

PACS numbers: 61.46.+w, 79.70.+q, 73.63.Fg

∗anthony.ayari@lpmcn.univ-lyon1.fr

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Nano-electro-mechanical systems (NEMS)[1] are under extensive research owing to their

potential for radio frequency communication and highly sensitive sensors. This research,

before becoming applicable, will have to cope with several major issues such as crosstalk.

Since the work of ref. 2, a new class of NEMS has been experimentally demonstrated that

could circumvent this drawback by nano-active feedback. In contrast to quartz-oscillator like

architecture,[3] there is no need for macroscopic external active circuit since the nanodevice

itself is placed in a self-oscillating regime. This concept was first theoretically proposed for

NEMS by Gorelik et al.[4] in the specific case of the charge shuttle and is now observed in

a large variety of experimental configurations[2, 5–9]. Although the work of ref. 2 reaches

qualitative agreement between experiment and modelling of the self-oscillation phenomenon,

it lacks simple arguments about the origin of the instability. Here, we derive a simple

linearized model and an equivalent purely electrical circuit that helps one getting further

insight on the way to design and scale down such an oscillator. This model is then validated

by dynamical and finite element simulations. The idea exposed in this article, with minor

adaptations, could be useful for other experimental geometries.

In a typical experiment, a nanowire (NW) or nanotube with resistance RNWis attached

to a tungsten tip in front of an anode connected to the ground [Fig. 1(a)]. The tip is

at a negative DC voltage −VDCfrom the ground; electrons are emitted from the apex of

the nanowire by field emission and collected by the anode. The NW starts to oscillates

spontaneously in the transverse direction when VDCis larger than some voltage threshold.

This system can be modeled by two coupled differential equations (see Eq. 1-2 in ref. 2):

first, a mechanical equation that can be linearized as follow:

¨ x +ω0

Q˙ x + ω2

0x = H¯UU,(1)

where x is the transverse displacement of the apex of the NW compared to the equilibrium

position (taken positive when the NW approaches the anode), 2πω0the resonance frequency

of the mechanical oscillator, Q the quality factor and H a positive parameter characterizing

the actuation strength by electrostatic forces between the wire and the anode. These pa-

rameters are supposed to be relatively constant in the range of interest.¯U is the DC voltage

between the NW and the anode and U the AC voltage.¯U is not equal to VDCas a result of

the voltage drop through the nanowire. Second, the linearized electrical equation reads:

?∂IFN

∂U

+

1

RNW

?

U + C˙U = −∂IFN

∂x

x − C′¯U ˙ x, (2)

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C

R Rm m

L Lm m

IFN

C Cm m

R RNW NW

VDC

U+U

i imot

IAC

W tipx x

e-

NW

Anode

(a)

(b)

FIG. 1: (Color online) (a) Schematic of the experimental configuration and (b) schematic of the

equivalent purely electrical circuit of the self-oscillation of the nano electro mechanical system of

ref. 2.

where C is the capacitance between the NW and the anode, C′its derivative with respect

to position, and IFN(U +¯U;x) the field emission current described by the Fowler–Nordheim

equation IFN= A(U +¯U)2β2exp(−B/(U +¯U)β). The x dependence of IFNcomes from the

field enhancement factor β.

An important point to notice is that the field emission characteristics depends on two

inputs, the apex voltage and its 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 degree of freedom x. A

simple equivalent electrical circuit is shown in Fig. 1(b). The electro-mechanical resonator

is represented by a series RLC circuit in parallel with the capacitor C of Eq. 2. In this

well-known analogy, the motional current through the RLC circuit is imot= C′¯U ˙ x and the

passive components are the motional inductance Lm= 1/(H¯U2C′), the motional resistance

Rm= ω0/(QH¯U2C′) and the motional capacitance Cm= H¯U2C′/ω2

0. The voltage across

the motional capacitance is proportional to x and can be used as the gate voltage of an

equivalent transistor delivering the same field emission current for a given x and U +¯U.

The transconductance of such transistor is (∂IFN/∂x)H¯U/ω2

0. It brings the gain necessary

to sustain the self-oscillation regime and acts as a feedback loop.

The main parameter of the self-oscillating circuit is the driving DC voltage above which

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the system spontaneously generates the AC signal. In the following, we derive a simple

analytical formula giving the self-oscillation condition. If the nanowire resistance RNWis

smaller than the field emission resistance (∂IFN/∂U)−1, to first order the voltage at the apex

¯U is VDCand there is no self-oscillation. We consider the opposite case RNW≫ (∂IFN/∂U)−1

because it gives a simpler formula (the general case can be calculated straightforwardly by the

same method). However, when the nanowire resistance gets larger more power is dissipated

in heating instead of sustaining the oscillation, so that it might seem optimal to keep RNW

larger than the field emission resistance by less than an order of magnitude.A single

differential equation of the full electro-mechanical system can be obtained by combining

Eqs. 1 and 2:

τ...

x + ¨ x

?

1 +ω0τ

Q

?

+ ˙ x

?ω0

Q+ H¯U2τ∂ lnC

∂x

?

+ ω2

0τ

?

+x

?

ω2

0+ H¯U2∂ lnβ

∂x

= 0(3)

where τ = C(∂IFN/∂U)−1is the discharge time constant of the electrical circuit. According

to the Routh–Hurwitz criterion this dynamical system is stable when:

H¯U2τ

?∂ lnβ

∂x

−∂ lnC

∂x

?

1 +ω0τ

Q

??

−ω0τ

Q

?1

τ+ τω2

0+ω0

Q

?

≥ 0(4)

From this inequality, since C and β increase with x, only the variation of β with x favors the

self-oscillation regime and we can distinguish between two categories of terms that prevent

from reaching it: i) the variation of the capacitance with x and ii) the relative value of τ and

ω−1

0. The latter can be minimized for ω0τ ∼ 1 as long as Q ≫ 1 (our nanowire resonators[10]

routinely reach Q > 105). In these conditions, the geometry of the device should be such

that ∂ ln(β/C)/∂x > 0 to have a chance to observe self-oscillations. Finally the threshold

DC voltage at the apex for self-oscillation is:

¯Uso=

ω0

?QH∂ ln(β/C)/∂x

(5)

and the threshold DC voltage of the power supply is VDC

so

=¯Uso+ RNWIFN(¯Uso,β).

In order to check the different hypotheses made, we performed numerical simulations

and determined the electrostatic force, capacitance and field enhancement factor by finite

element methods (FEM). The sample is a straight 10 µm-long nanowire of radius 100 nm

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v

r

11.522.5

10

−1

10

0

10

1

10

2

5

10

15

x 10

−5

λ

FIG. 2: (Color online) Stability map of a nanowire during field emission for Q = 10000 and different

normalized voltages v and dimensionless intrinsic frequencies r.

attached to a metallic conical tip in front of a metallic plate perpendicular to the axis of the

tip. The nanowire is initially tilted by 20◦compared to the cone axis. The sole degree of

freedom of the nanowire is this angle that can decrease due the attractive electrostatic force

between the wire and the metallic plate. The distance between the tip end and the plate is

60 µm. The mechanical restoring force is taken from the calculated rigidity of a clamped

free beam with a Young modulus of 400 GPa and density of 3200 kg/m3, Q = 104and

RNW= 1010Ω. Further details about the simulations and a more refined mechanical model

can be found in ref. 11. We first simulated the spatial variation of C and β and verified

that ∂ ln(β/C)/∂x > 0 for a wide range of angles around 20◦, and established that H is

changing by less than 15%. The dimensionless differential equations were then rewritten,

their eigenvalues computed, and the sign of their real part λ scrutinized. The real part defines

the growth rate of the mode and the solution, which is proportional to exp(λt), decay to zero

when it is negative, so that the system is stable. On the contrary, λ > 0 makes the system

unstable and leads it into a stable self-oscillating regime thanks to nonlinear saturating

terms. The oscillation amplitude gets larger as λ increases. Finally, we determined stability

maps giving the parameter regions where λ is positive and self-oscillations possible.

Fig. 2 represents the stability map of the system for different applied DC voltages v =

VDC/Vrefand different dimensionless intrinsic frequencies r = ω0τ. Vref= 400V is the voltage

above which RNWstops being negligible when compared to the field emission resistance. One

can point out that i) there is no self-oscillation for v ≪ 1, ii) self-oscillations are easier at

higher v (the growth rate is larger and the instability region wider). This validates the

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1 1.522.5

0

1

2

3

4

5x 10

4

v

Q

0.5

1

1.5

2

2.5

3

x 10

−4

λ

FIG. 3: (Color online) Stability map of a nanowire during field emission for a dimensionless fre-

quency r = 5 and different normalized voltages v and Q. The solid line represents the self-oscillation

threshold determined using Eq. 5.

statement that for optimal self-oscillations RNWneeds to be bigger than the field emission

resistance (the field emission current increases exponentially with v, so that the field emission

resistance is smaller for 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 quality factors. Eq. 5 that determines

the boundary between the stable region and the self-oscillation region is in relatively good

agreement with the results of numerical simulations for high voltage, i.e. when ∂IFN/∂U ≫

1/RNW. This confirms the validity of the above analytical derivation. Eq. 5 shows that,

as for any other NEMS device, keeping good performance (in this case by maintaining the

operating voltage low) at the nanoscale and high frequency requires an improvement of the

capacitive coupling and the quality factor. Finally a simple scaling calculation shows that r

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 the nanowire

or nanotube saturates in the ballistic regime, the device can still be operated with the help

of an additional constant resistance between the DC power supply and RNW.

In conclusion, using an electrical equivalent circuit, we showed that the origin of self-

oscillations in field emission NEMS can be understood in terms of motional capacitance

and spatial variation of the field emission current in a feedback loop. An equation was

derived to determine the threshold voltage for self-oscillation and its output confirmed by

numerical and FEM simulations.We expect that our simple model will demystify the

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mechanism responsible for self-oscillation in field emission NEMS, as it appears that it can

be understood with simple classical electrical passive components and one transistor. It

appears then that geometries like the one of ref. 6 where the self-oscillation mechanism is

not yet clearly identified are indeed very similar to ours and may be understood within the

same framework. This work opens up perspectives for the control and fabrication of low

power nano-oscillators for time base and AC generators applications.

Acknowledgments

This work was supported by French National Research Agency (NEXTNEMS : ANR-07-

NANO-008-01 and AUTONOME : ANR-07-JCJC-0145-01) and R´ egion Rhˆ one-Alpes CIBLE

program. The authors acknowledge the “plateforme nanofils et nanotubes lyonnaise”.

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[3] E. Colinet, L. Duraffourg, S. Labarthe, P. Andreucci, S. Hentz, and P. Robert, Journal of

Applied Physics 105, 124908 (2009).

[4] L. Y. Gorelik, A. Isacsson, M. V. Voinova, B. Kasemo, R. I. Shekhter, and M. Jonson, Phys.

Rev. Lett. 80, 4526 (1998).

[5] H. Kim, H. Qin, and B. R. H., New J. Phys. 12, 033008 (2010).

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[11] A. Lazarus, E. de Langre, P. Manneville, P. Vincent, S. Perisanu, A. Ayari, and S. Purcell,

Int. J. Mech. Sci. p. in press (2010).

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