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arXiv:0811.0870v2 [cond-mat.mes-hall] 18 Nov 2008
Nonlinear Dynamics and Chaos in Two Coupled Nanomechanical
Resonators
R. B. Karabalin, M. C. Cross, and M. L. Roukes
Department of Physics and Kavli Nanoscience Institute,
California Institute of Technology, Pasadena CA 91125
(Dated: October 24, 2018)
Two elastically coupled nanomechanical resonators driven
independently near their resonance
frequencies show intricate nonlinear dynamics. The dynami
cs provide a scheme for realizing a
nanomechanical system with tunable frequency and nonlinea
r properties. For large vibration ampli-
tudes the system develops spontaneous oscillations of ampl
itude modulation that also show period
doubling transitions and chaos. The complex nonlinear dyna
mics are quantitatively predicted by a
simple theoretical model.
PACS numbers: 05.45.-a, 62.25.-g, 85.85.+j
Resonant nanoelectromechanical systems (NEMS) [1]
are attracting interest in a broad variety of research areas
and for many possible applications due to their remark-
able combination of properties: small mass, high operat-
ing frequency, large quality factor, and easily accessible
nonlinearity [2]. Further development of NEMS applica-
tions such as superior mass [3], force [4] and charge [5]
sensors, or reaching the quantum limit of detection in me-
chanical systems [6], requires addressing several impor-
tant challenges. For example, the nonlinearity of the de-
vices has to be either minimized or utilized for improving
performance [7]. In addition, the large-scale integration
of nanodevices demands a detailed understanding of the
behavior of coupled devices in NEMS arrays [8, 9, 10, 11].
In this paper we demonstrate complex nonlinear be-
havior of a pair of coupled nanomechanical devices, and
show that this can be quantitatively understood from
the basic physics of the devices. We show that the linear
and weakly nonlinear response of one oscillation can be
modified by driving the second oscillation, and, for some
ranges of parameters of the devices, that the linear re-
sponse range of the first oscillation can be significantly
extended. When both oscillations are driven into their
strongly nonlinear range more complicated frequency-
sweep response curves are found, corresponding to the
well known bistability of driven anharmonic “Duffing”
resonators, but now with switching between a variety
of different stable states of the coupled pair. Sponta-
neous amplitude modulation oscillations may develop,
with frequencies characteristic of the dissipation rates
rather than of the intrinsic frequencies or their sums and
differences. These amplitude modulations show period
doubling bifurcations and chaos. The complex dynam-
ics are reproduced quantitatively by a simple theoretical
model, giving us confidence that the nonlinear behav-
ior of coupled nanomechanical devices can be understood
and controlled.
We study a system of two strongly coupled nonlin-
ear nanoelectromechanical resonators using a structure
of doubly-clamped beams with a shared mechanical ledge
shown in Fig. 1a. The devices consist of a stack of three
layers of gallium arsenide (GaAs): a 100nm highly n-
doped layer, a 50nm insulating layer, and another 50nm
layer that is highly p-doped. The piezoelectric property
of GaAs results in a highly efficient integrated actuation
mechanism described in [12]. A preliminary 120nm deep
etch step is done to isolate the actuation electrodes of the
two beams, so that the two beams can be addressed sep-
arately while retaining strong elastic coupling. Optical
interferometry is used for the motion transduction [13].
The laser beam is adjusted so that both beams are in the
illuminated spot.
We model the behavior of the two strongly interacting
nonlinear resonators by a system of coupled equations of
motion for the beam displacements
x
1
, x
2
in their funda-
mental modes
̈
x
1
+
γ
1
̇
x
1
+
ω
2
1
x
1
+
α
1
x
3
1
+
D
(
x
1
x
2
) =
g
D
1
(
t
)
,
(1a)
̈
x
2
+
γ
2
̇
x
2
+
ω
2
2
x
2
+
α
2
x
3
2
+
D
(
x
2
x
1
) =
g
D
2
(
t
)
.
(1b)
As well as the usual terms describing the resonant fre-
quencies, damping, and Duffing nonlinearity, we include
a linear coupling term in the displacements of strength
D
. The terms on the right hand side are the external
drives applied to the two beams, which are controlled in-
dependently. The linear terms in the equations, ignoring
for now the drive and dissipation, give two modes with
frequencies
ω
I
and
ω
II
and corresponding eigenvectors
e
I
,
e
II
[14]. The frequency difference of the modes re-
sults from both the intrinsic frequency difference
ω
1
ω
2
and the coupling
D
.
To investigate the behavior of the system, the two
beams are connected to two different sources with in-
dependent frequencies and amplitudes. The monitored
output variables are the amplitudes and phases of a lin-
ear combination of the mechanical displacements of the
two beams at or near the two drive frequencies [21]. By
applying various combinations of small amplitude sig-
nals to the two beams at frequencies near the mode res-
onances, the linear coupling parameters can be deter-
mined. For the particular device shown in Fig. 1a the
mode frequencies are determined to be
ω
I
/
2
π
= 16
.
79
MHz and
ω
I
/
2
π
= 17
.
25 MHz and the eigenvectors
2
FIG. 1: (color online) (a) SEM image of the system (beam
dimensions: 6
μ
m x 500nm x 200nm). (b) Up (dark, blue) and
down (light, yellow) frequency sweeps of the amplitude
|
A
|
of
the response for a single drive of strength 4.3 times critica
l
as a function of the frequency relative to the linear resonan
ce
frequency
ω
0
and scaled by the width
γ
. The amplitude is
plotted in units of the maximum amplitude at the critical
drive strength. (c) Frequency shift of the weakly driven firs
t
mode as a function of the displacement amplitude
|
A
II
|
of the
strongly driven second mode: points and solid line – experi-
ment; dashed line – small amplitude theory.
e
I
= (0
.
854
,
0
.
521) and
e
II
= (
0
.
521
,
0
.
854). The fre-
quencies of the individual resonators determined from in-
verting the mode equations are
ω
1
/
2
π
= 16
.
71 MHz and
ω
2
/
2
π
= 16
.
92 MHz. The frequency separation of 200
kHz is consistent with the fabrication tolerance. The
coupling strength
D/
2
π
= 2
.
63 MHz is consistent with
the strength of the elastic coupling found by finite ele-
ment simulations of similar devices [15]. Transforming
the measured width of the modes back to the original
equations determines the values of
γ
1
, γ
2
. The nonlinear
parameters
α
1
, α
2
are deduced from the expression for
the geometric nonlinearity [7] using the beam thickness
known from the fabrication and lengths calculated from
the beam frequencies
ω
1
, ω
2
and the material constants
[22].
The response of the system driven near resonance
and for small dissipation and driving can be calculated
from (1) using the standard methods of secular pertur-
bation theory [2]. This approach has previously used
for the case of parametrically driven nanomechanical de-
vices [9, 10, 11], where hysteretic switches between dif-
ferent stable states in frequency sweeps were also pre-
dicted. Briefly, we introduce the slowly varying com-
plex mode amplitudes
A
I
, A
II
and forces
F
I
, F
II
using
x
[
I
]
= Re(
A
[
I
]
e
[
I
]
t
) and
g
D
[
I
]
= Re(
F
[
I
]
(
t
)
e
[
I
]
t
) (where
[
I
] stands for either
I
or
II
), substitute into the equa-
tions of motion, and retain only the near resonant terms.
This reduces the equations of motion to
2
I
̇
A
I
+
I
γ
I
A
I
+
α
I
|
A
I
|
2
A
I
+
β
I
|
A
II
|
2
A
I
=
F
I
(
t
)
,
(2)
and a corresponding equation for
A
II
. The mode nonlin-
earity parameters
α
[
I
]
and
β
[
I
]
are calculated from
α
1
, α
2
and the eigenvectors
e
I
,
e
II
so that all the parameters in
(2) are known from linear measurements and the beam
geometry and material constants.
We first look at the case where each mode responds at
the drive frequency
ω
D
[
I
]
which is set near the resonant
frequency
ω
[
I
]
so that
A
[
I
]
e
i
(
ω
D
[
I
]
ω
[
I
]
)
t
, with
|
A
I
|
2
=
|
F
I
|
2
[2
ω
I
(
ω
DI
ω
I
)
α
I
|
A
I
|
2
β
I
|
A
II
|
2
]
2
+
ω
2
I
γ
2
I
,
(3)
etc. For a single drive (e.g.
A
II
=
F
II
= 0), so that
the cross-mode nonlinear coupling proportional to
β
[
I
]
is not involved, this expression reproduces the regular
Duffing response curve [16]. Prominent features are the
shift of the frequency of the maximum response to larger
values for positive
α
(nonlinear spring stiffening), and
bistability and hysteresis that develop above a critical
drive strength. An experimental example of upward and
downward frequency sweeps for a drive strength 4.3 times
critical is shown in Fig. 1b.
FIG. 2: (color online) Frequency sweeps of the first mode
response for increasing amplitudes of the second mode: the
dark (blue) lines are for upward sweeps; the light (yellow)
line in (a) is a downward sweep, showing the hysteresis and
amplitude jumps in this case. There are no hysteretic jumps
for (b) and (c), and so the downward sweeps are not shown.
We first look at the linear and weakly nonlinear re-
sponse of the first mode when the second mode is driven
into its strongly nonlinear regime. Retaining the lead-
ing order effect of the first mode on the intensity of the
second mode we can write the numerator in (3) as
[
2
ω
I
(
ω
DI
ω
I
)
β
I
|
A
(0)
II
|
2
̄
α
I
|
A
I
|
2
]
2
+
ω
2
I
γ
2
I
,
(4)
where
A
(0)
II
is the solution for the second mode in the
absence of mode I given by the equation for
A
II
corre-
sponding to (3) assuming zero
A
I
. The new parameter
̄
α
I
is an effective Duffing nonlinearity coefficient and is
given by ̄
α
I
=
α
I
2
β
I
β
II
ω
II
|
A
(0)
II
|
2
/∂ω
DII
. Equation
(4) predicts two important effects: the frequency tun-
ing of the first mode proportional to the square of the
amplitude of the second mode (upwards for positive
β
I
);
and the change in the effective nonlinear coefficient for
3
the motion of the first mode depending on the excitation
strength of the second mode through the last term in ̄
α
I
.
To test the frequency tuning we excite the second mode
at a drive level approximately 4.3 times the critical value
so that the spectral response is the strongly nonlinear
Duffing curve (see Fig. 1b). As the actuation frequency
of the second mode is steadily increased in small steps
its vibration amplitude rises over a wide frequency range
until it drops to the lower amplitude state beyond the
maximum. The evolution of the spectral response of the
first mode is monitored at a driving level approximately
four times lower than the critical value for this mode us-
ing a network analyzer. The dependence of the first mode
frequency shift on the vibration amplitude of the second
mode is shown in Fig. 1c. The experimental results for
frequency tuning closely follow the predicted parabolic
dependence
β
I
|
A
II
|
2
for amplitudes up to about 20nm.
Monitoring the frequency shift of one mode proportional
to the square of the displacement of a second mode has
been proposed for quantum nondemolition measurements
in nanoelectromechanical systems [17, 18].
If the actuation level of the first mode is increased
above the onset of nonlinearity, while the second mode
drive is kept at a much higher level, then the effec-
tive nonlinear coefficient ̄
α
I
is decreased. The decrease
is largest when the second mode is driven on the por-
tion of the Duffing response curve where the intensity
is increasing linearly with the drive frequency
|
A
II
|
2
2
ω
II
(
ω
DII
ω
II
)
II
. This gives the minimum effective
nonlinear coefficient ̄
α
I,
min
=
α
I
(
1
β
I
β
II
α
I
α
II
)
. This re-
sult indicates that if the coupling is strong enough, so
that
β
I
β
II
> α
I
α
II
, the minimum value of the effective
nonlinear coefficient is negative. In this case the reso-
nance curve tilts to the left, as opposed to the usual case
for a doubly clamped beam where the peak leans to the
right. It also means that the nonlinear coefficient van-
ishes for some drive strength and frequency of the second
mode.
Some experimental results for the first mode driven
at twice the critical strength illustrating this effect are
shown in Fig. 2. The plots show the shape of the res-
onance peak for three values of the drive frequency of
the second mode. In panel (a), the amplitude
A
II
of the
second mode is low and the first mode spectral response
has the regular nonlinear Duffing shape leaning to the
right. For larger
A
II
as in (b), the first mode resonance
peak shape assumes a form close to a Lorentzian with
little nonlinearity apparent. For even larger
A
II
as in
(c) the sign of the effective Duffing coefficient becomes
slightly negative, causing the spectral response peak to
lean to the left. The quenching of the nonlinearity in (b)
could be used to enhance the limited dynamic range [7]
of nanomechanical devices.
If the drive level of the first mode is increased further
a variety of new effects can be observed. Under these
FIG. 3: (color online) First mode frequency response as in
Fig. 2 for increasing values of the second mode drive frequen
cy
but for stronger driving of the first mode (both modes are
driven at about four times the critical strength). The top
plots are experimental measurements while the bottom ones
are theoretical simulations. The second mode response was
monitored, but is not shown.
conditions the dynamics of the system is not fully ex-
plained by the steady state solutions as in (3), and so
we solve for the expected behavior by numerically inte-
grating the time-dependent coupled equations (2). Now
the behavior is more complex, as the response of the first
mode becomes large enough to cause transitions in the
second mode response through the nonlinear coupling.
As a result the spectral response curves acquire peculiar
nontrivial shapes, as shown in Fig. 3. Numerical sim-
ulations of the equations give good predictions for the
complex phenomena.
An obvious difference between the experimental and
theoretical plots in Fig. 3 is the noisy regions on the theo-
retical curves near the up and down transitions. This dif-
ference actually results from the different ways the plots
are generated in theory and experiment, and a more care-
ful investigations shows consistent and interesting dy-
namics in both experiment and theory: for the drive
frequencies near the transition points the fast (about
17MHz) oscillating response becomes amplitude modu-
lated with a frequency of about 10 to 20 kHz [23]. The
theoretical equations (2) show that the frequency for the
amplitude modulation is determined by the line width,
which in our experimental setup is about 8 kHz, and is
not related to sum and difference frequencies of the two
modes.
Examples of simulated and measured amplitude mod-
ulation dynamics for three different sets of drive parame-
ters are shown in Fig. 4 for a second device with different
parameters to the one used for Figs. 1-3. The first column
shows examples of numerically calculated phase portraits
of Re
A
II
versus Re
A
I
obtained by solving the time de-
4
FIG. 4: Complex dynamics of two strongly coupled nanome-
chanical resonators. The three rows correspond to different
input parameters (drive frequencies and amplitudes). The
first column shows the theoretical calculation of the phase
portrait, the second shows its experimental measurement, a
nd
the third column shows the corresponding experimental powe
r
spectrum of the optical measurement near one of the drive
frequencies.
pendent equations (2). Phase portraits measured experi-
mentally are shown in the second column [24]. These are
obtained using homodyne down-conversion of the trans-
duced mechanical signals from both modes, which are
then read by independent oscilloscope channels. We also
perform wider frequency band spectrum analyzer mea-
surements shown in the third column. Since the nature
of the dynamics is sensitive to the precise values of the
system parameters, the drive strength and frequencies
in the experiment are slightly adjusted from the values
corresponding to the theoretical plots to produce compa-
rable phase portraits.
The top row of Fig. 4 shows a relatively simple example
where the motion is periodic but the
A
I
A
II
trajectory
forms a small figure-of-eight loop. The dynamics can be
roughly understood in terms of transitions between states
with the first mode at large amplitude and the second
mode at small amplitude and
vice versa
. The spectrum
of the measured mechanical signal shows satellite peaks
corresponding to the anharmonic amplitude modulation.
As the parameters of the system are changed, we have
observed period doubling or quadrupling in the ampli-
tude modulation, where the phase trajectory takes two
or four revolutions in order to complete the cycle [19].
An example of period quadrupling is shown in the mid-
dle row of Fig. 4. The complicated loop structures are
visible in both theoretical and experimental trajectories,
and the spectrum reveals amplitude modulation peaks
at frequencies that correspond to double and quadruple
periods. Period doubling transitions are often associated
with chaotic dynamics [19], and indeed for other param-
eter values we observe chaos in our coupled nanoelec-
tromechanical system and in the theoretical model, as
shown in the bottom row of Fig. 4. The evidence for the
chaotic dynamics in the experiment is the broad band
component to the spectrum (evident in the shoulders to
the amplitude modulation peaks), and a phase portrait
trajectory that does not form a closed loop [25]. The
theoretical model shows a similar phase portrait.
Our detailed study of two elastically coupled, indepen-
dently driven, nanomechanical beam resonators reveals
complex nonlinear dynamics with a number of potential
applications. For example, driving one of the modes can
be used to tune the effective nonlinearity of the other
mode. This can be used to significantly increase the dy-
namic range of the resonator by quenching the effective
nonlinearity. In a first approximation, the motion of one
mode couples quadratically to the resonance frequency of
the other mode, a phenomenon that has been proposed
for quantum nondemolition measurements in nanome-
chanical systems. For larger vibration amplitudes spon-
taneous oscillations of amplitude modulation develop, at
a frequency determined by the resonator ring-down time.
These oscillations show period doubling and chaos char-
acteristic of strongly nonlinear systems. The full range
of complex dynamics investigated is quantitatively re-
produced by theory. Our success at predicting and sub-
sequently observing quite delicate features of the non-
linear dynamics is strong evidence that the nonlinearity
and coupling in arrays of nanomechanical devices can be
quantitatively understood and controlled.
Many of the ideas leading to this work were devel-
oped in collaboration with Ron Lifshitz supported by the
U.S.-Israel Binational Science Foundation (BSF) through
Grant No. 2004339. We thank Matt Matheny for many
useful discussions.
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[21] The combination measured depends on the geometry of
the optical spot relative to the beams, which we deter-
mine from the linear experiments using a variety of drive
combinations.
[22] The structure of the dynamical system depends on the
ratio
α
I
II
which is close to unity for our nearly identi-
cal beams. The absolute magnitude of
α
(approximately
0.0072 (MHz/nm)
2
) calibrates the amplitude of the re-
sponse in nm.
[23] The theoretical points, which are the end values of a lon
g
numerical simulation, depend on the phase of the ampli-
tude modulation at the end of each run, whereas this
modulation leads to a drop in the experimentally mea-
sured amplitude since RF power is transferred outside
the measurement bandwidth.
[24] The full phase space is four dimensional,
Re
A
I
,
Re
A
II
,
Im
A
I
,
Im
A
II
; we plot a two dimen-
sional projection.
[25] Previous experiments that suggested chaos in a nanome-
chanical system [20] demonstrated a complex behavior
of the response at one of the drive frequencies as that
parameter was swept, rather than complex dynamics for
fixed system parameters as shown in our work.