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)
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
Nano-electro-mechanical systems (NEMS) 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, 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. 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:
U + C˙U = −∂IFN
x − C′¯U ˙ x, (2)
R Rm m
L Lm m
C Cm m
W tip x x
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
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
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:
x + ¨ x
+ ˙ x
Q+ H¯U2τ∂ lnC
0+ H¯U2∂ lnβ
= 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:
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
0. The latter can be minimized for ω0τ ∼ 1 as long as Q ≫ 1 (our nanowire resonators
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:
and the threshold DC voltage of the power supply is VDC
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
1 1.52 2.5
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
1 1.52 2.5
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
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.
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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