Tunable, broadband nonlinear nanomechanical resonator.

Abstract

A nanomechanical resonator incorporating intrinsically geometric nonlinearity and operated in a highly nonlinear regime is modeled and developed. The nanoresonator is capable of extreme broadband resonance, with tunable resonance bandwidth up to many times its natural frequency. Its resonance bandwidth and drop frequency (the upper jump-down frequency) are found to be very sensitive to added mass and energy dissipation due to damping. We demonstrate a prototype nonlinear mechanical nanoresonator integrating a doubly clamped carbon nanotube and show its broadband resonance over tens of MHz (over 3 times its natural resonance frequency) and its sensitivity to femtogram added mass at room temperature.

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Tunable, Broadband Nonlinear
Nanomechanical Resonator
Hanna Cho, Min-Feng Yu,* and Alexander F. Vakakis*
Department of Mechanical Science and Engineering, University of Illinois at UrbanasChampaign, 1206 West Green
Street, Urbana, Illinois 61801
Lawrence A. Bergman, and D. Michael McFarland
Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 South Wright Street,
Urbana, Illinois 61801
ABSTRACT A nanomechanical resonator incorporating intrinsically geometric nonlinearity and operated in a highly nonlinear regime
is modeled and developed. The nanoresonator is capable of extreme broadband resonance, with tunable resonance bandwidth up to
many times its natural frequency. Its resonance bandwidth and drop frequency (the upper jump-down frequency) are found to be
verysensitivetoaddedmassandenergydissipationduetodamping.Wedemonstrateaprototypenonlinearmechanicalnanoresonator
integratingadoublyclampedcarbonnanotubeandshowitsbroadbandresonanceovertensofMHz(over3timesitsnaturalresonance
frequency) and its sensitivity to femtogram added mass at room temperature.
KEYWORDS Nonlinear resonance, nanoresonator, broadband resonance, responsivity, carbon nanotube
R
physicalquantities1-12andevenquantuminteractions.13-15
However,thereduceddevicesizereducesitsdynamicrange
(down to the nanometer scale) for linear operation,16which
makes developing the required measurement system dif-
ficult and accordingly limits its sensitivity, especially under
ambient and room-temperature environments.
The main element in most mechanical nanoresonators
consistsofananoscalemechanicalcantileverorananoscale
doubly clamped beam, which significantly reduces the ef-
fective mass of the resonance system. A general feature of
suchdevicesisthattheyoperatepredominantlyinthelinear
regime and achieve high sensitivity to mass or charge
through the realization of high quality-factor resonance at
high frequency. Most noticeably, their recent development
has allowed the sensing of mass down to the zeptogram (zg)
level,7,12for even a single molecule,9,11and the transport
of a single electron charge.17,18Whereas the absolute mag-
nitude of the involved resonance amplitude is small, the
relative magnitude is actually quite significant when com-
pared to the reduced device size. As a result, such nanoscale
resonance systems can easily transition from linear reso-
nance operation to a nonlinear one through a slight increase
in its dynamic operating amplitude.16-18The importance of
nonlinearity in such nanomechanical resonance systems is
ecent advances have seen the development of na-
nomechanical resonators operating in the linear
regimethatarecapableofdetectingextremelysmall
thus gaining more recognition. For example, electrostatic
interactions19and coupled nanomechanical resonators20
were proposed for tuning the nonlinearity in nanoscale
resonance systems; noise-enabled transitions in a nonlinear
resonator were analyzed to improve the precision in mea-
suring the linear resonance frequency21and a homodyne
measurement scheme for a nonlinear resonator was pro-
posed for increasing the mass sensitivity and reducing the
response time.22In addition, the basins of attraction of
stable attractors in the dynamics of a nanowire-based me-
chanical resonator were studied,23and the nonlinear behav-
ior of an embedded24and a curved carbon nanotube25was
theoretically investigated. Such studies increasingly offer a
new conceptual understanding and thus strategies to deal
with and even exploit the increasingly prominent nonlinear
behaviorinthedevelopmentofnanomechanicalresonators.
Herein, we design an intrinsically nonlinear nanome-
chanical resonator defined by the inherent geometric non-
linearity that can be readily incorporated into practical
device development, and we apply both theoretical model-
ing and experimental validation to demonstrate its tunabil-
ity, its capacity for broadband resonance, and its sensitivity
to added mass and to energy dissipation due to damping.
The intrinsic nonlinearity is simply introduced into the
nanoscale resonance system through a geometric design as
described in the following. Consider a fixed-fixed mechan-
ical resonator employing a linearly elastic wire with negli-
gible bending stiffness and no initial axial pretension. When
driven transversely by a periodic excitation force applied
locally to the middle of the wire, it will exhibit strong
geometric nonlinearity and become an intrinsically (purely)
*Corresponding authors. E-mail: mfyu@illinois.edu and avakakis@illinois.edu.
Received for review: 02/09/2010
Published on Web: 04/12/2010
pubs.acs.org/NanoLett
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nonlinear resonator (Figure 1A). In such a resonator, the
force-displacementdependenceisdescribedbytherelation
F ) kx[1 - L(L2+ x2)-1/2] ≈ (k/2L2)x3+ O(x5)26, where F is
the transverse point force applied to the middle of the wire,
x is the transverse displacement in the middle of the wire,
and L and k are the half-length and the effective axial spring
constant of the wire, respectively. The total absence of a
linear force-displacement dependence term (i.e., a term of
the form kx) results in the realization of a geometrically
nonlinear force-displacement dependence of pure cubic
order. This resonator has no preferential resonance fre-
quency, and its resonance response is broadband,26which
is conceptually different from typical linear mechanical
resonators. Moreover, the apparent resonance frequency is
completely tunable through the instantaneous energy of the
system. If the bending effects are non-negligible or if an
initial pretension exists in the wire, then a nonzero linear
term in the previous force-displacement relation is in-
cluded, giving rise to a preferential resonance frequency.
However,aslongasthispreferentialfrequencyissufficiently
small compared to the frequency range of the nonlinear
resonance dynamics, the previous conclusions still apply.
Thus,weproceedtoanalyzeadoublyclampedEuler-Bernoulli
beam having a foreign mass (mc) attached to its middle and
excited transversely by an alternating center-concentrated
force. Considering the geometric nonlinearity induced by
axial tension during oscillation, the vibration of the beam is
described by
where w(x, t) is the transverse displacement of the beam
with x and t denoting the spatially and temporally indepen-
dent variables, E and F are the Young’s modulus and mass
density, A and L are the cross-sectional area and half length
of the beam, I is the area moment of inertia of the beam, Q
is the quality factor of the resonator in the linear dynamic
regime, F is the excitation force applied to the middle of the
beam, ω() 2πf) is the driving frequency, and ωo() 2πfo) is
the linearized natural resonance frequency of the beam. It
isassumedthatnoinitialaxialtensionexistswhenthebeam
is at rest, and shorthand notation for partial differentiation
is used.
The transverse displacement of the beam can be ap-
proximately expressed as w(x, t) ) ∑i ) 1
Wi(x) is the ith mode shape of the linearized beam, φi(t) is
thecorrespondingithmodalamplitude,andNisthenumber
of beam modes considered in the approximation. The
leading modal amplitude, φ1(t), is then approximately gov-
erned by a Duffing equation obtained by discretizing eq 1
through a standard one-mode Galerkin approach27(Support-
ing Information):
N
Wi(x) φi(t), where
Here, M ) [mc/(2FAL)]W12(L) ) (mc/m0)W12(L) is the ratio of
the foreign mass to the overall mass of the beam multiplied
by a factor due to the center-concentrated geometry of the
foreign mass distribution (when the foreign mass is distrib-
uted evenly on the beam, M ) mc/m0); the amplitude of the
driving force per unit mass in eq 2 is defined by q ) W1(L)F/
mo, and the nonlinear coefficient is defined by
Following a harmonic balance approximation27with a
single frequency ω, we find that the response spectrum of
FIGURE 1. Nanoresonator integrating intrinsic geometric nonlin-
earity. (A) Schematic showing a simple doubly clamped mechanical
beam(anditsequivalentspringmodel)havinganintrinsicgeometric
nonlinearity. (B) Tunability of the resonance bandwidth of a non-
linear nanoresonator. The plot shows the dependence of the drop
frequency/natural frequency ratio on the applied drive force and the
quality factor of the mechanical resonator. The plot in the inset
shows the frequency response of a nonlinear resonator calculated
on the basis of the parameters listed for carbon nanotube B1 in the
inset of Figure 2.
[FA + mcδ(x - L)]wtt+ (mωo/Q)wt+ EIwxxxx-
(EA/4L)wxx∫0
2Lwx
2dx ) F cos ωtδ(x - L) (1)
(1 + M)φ1
••
+
ω0
Qφ1
•
+ ω0
2φ1+ αφ1
3) q cos(ωt)
(2)
α ) -
E
32FL4∫0
2LW1W1′′dx∫0
2L(W1′)2dx
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this Duffing oscillator forms a multivalued region when the
oscillation amplitude is over a critical value as seen in the
inset of Figure 1B. Specifically, there are two branches of
stableresonancesthatareconnectedbyabranchofunstable
resonances. As the frequency is swept up, the resonance
amplitude in the upper branch of stable resonances in-
creases up to the maximum possible amplitude and then
drops abruptly to a lower value as the forced motion makes
a transition to the lower stable branch. The drop frequency,
fdrop, at which this jump phenomenon occurs is approxi-
mately determined by the intersection of the Duffing re-
sponse spectrum with the free oscillation or the “backbone”
curve,27and its ratio to the linearized natural frequency is
given by
where Γ ) γ((FQ)/(E))2((2L)/(D))6((1)/(D4)) and γ ) 0.0303.
From this equation, it is clear that the drop frequency of this
nonlinearresonatorstronglydependsontheattachedcenter
mass and damping, besides the geometry of the beam and
the applied excitation force. A similar computation can be
performed for the reverse jump-up frequency during a
decreasing frequency sweep; in that case, the dynamics
follows a transition from the lower stable resonance branch
to the upper.
We estimate the mass responsivity (Rm), defined as the
shift in drop frequency with respect to the change in the
added center mass, as
Compared with a mass sensor based on a linear resonator
for which the responsivity is -fo/2mo, the nonlinear resona-
torutilizingthedropfrequencyasthemeasuranthasabetter
responsivity by a factor of rdrop[1 -(rdrop2- 1)/(2rdrop2- 1)]
whenignoringthetermW12(L)andifrdropg1.618.Themass
responsivitiesforthreerepresentativedoublyclampedbeams
with E ) 100 GPa and F ) 2600 kg/m3and a single-walled
CNT beam with E ) 1 TPa, for which the parameters are
listedintheinsettable,areplottedinFigure2Aasafunction
of the normalized frequency fdrop/fo. The value at fdrop/fo) 1
indicatestheresponsivityofalinearresonator.Itisapparent
that the responsivity is enhanced not only by considering a
nonlinear resonator with a smaller intrinsic mass and a
higher resonance frequency but also by increasing the ratio
ofthedropfrequencyoverthenaturalresonancefrequency.
This means that the performance of a mass sensor based
on a nonlinear nanoresonator can be considerably raised by
increasing its resonance bandwidth which, as we will show
later, is practically tunable.
For a nonlinear resonator to have such intrinsically
nonlinear behavior and a highly broadband resonance
response, several parameters, including the quality factor,
the size of the mechanical beam, and the driving force, are
to be optimized to provide a larger value of Γ according to
eq 3. Here, it is noted that the resonance bandwidth can be
extended by simply increasing the excitation force while
FIGURE 2. Sensing performance of a nonlinear nanoresonator to mass and energy dissipation due to damping. (A) Mass responsivities of four
different doubly clamped beams as a function of the drop frequency/natural frequency ratio. (B) Shift in the drop frequency for a 1% change
in the damping coefficient as a function of the drop frequency/natural frequency ratio. The inset table lists the parameters for the carbon
nanotubes used in the calculation.
rdrop)
fdrop
fo
)(1 + √1 + (1 + M)Γ
(1 + M)
)
1/2
(3)
Rm) lim∆mcf0
∆fdrop
∆mc
) -
fo
2mordrop(1 -
rdrop
2rdrop
2- 1
2- 1)W1
2(L) (4)
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keeping all other parameters of the resonator fixed. Figure
1B shows the tunability of the bandwidth up to 2 orders of
magnitude by simply changing the excitation force applied
to a nonlinear mechanical nanoresonator.
In addition, the drop frequency of the nonlinear nan-
oresonator is very sensitive to the magnitude of damping
associated with the resonance system under various ambi-
ent conditions, according to eq 3. The damping responsivity
of the drop frequency is estimated according to the change
in the damping coefficient ?, where ? ) 1/(2Q):
The shift in drop frequency for a 1% change in the damping
coefficient is plotted in Figure 2B and again shows the much
enhanced sensitivity offered by the intrinsically nonlinear
nanoresonator compared to that offered by the linear one.
We fabricated a nonlinear nanoresonator using a doubly
clamped carbon nanotube (CNT), of which a scanning
electron microscope (SEM) image is displayed in Figure 3A.
The device was fabricated through micromachining and
nanomanipulation. A silicon (100) wafer was coated with a
500-nm-thick silicon nitride layer followed by 1.5-µm-thick
silicon dioxide. A thin Cr/Au layer was then sputter coated
onto the silicon wafer and subsequently patterned through
photolithography to form a three-electrode layout. This
silicon wafer was back etched in KOH to make a thin
membrane of silicon dioxide under the electrodes. The
window was then milled with a focused ion beam to create
three suspended electrodes. Three vertical platinum posts
werefabricatedontothesethreeelectrodesthroughelectron-
beam-induced deposition. A high -quality multiwalled CNT
produced with an arc discharge was then selected and
manipulated inside an electron microscope and suspended
betweentwooftheplatinumpostswithbothendsfixedwith
electron-beam-induced deposition of a small amount of
platinum. The remaining platinum post was used as the
driving electrode for applying a localized oscillating electric
field to drive the oscillation of the CNT. The overall design
of the device maximized the localization of the excitation
force applied to the CNT beam (Supporting Information).
According to the previous discussion, the localization of the
applied force is necessary to create the strong geometric
nonlinearity in the resonance system.
To acquire the response spectrum of the nanoresonator,
the frequency of the applied ac driving voltage (Vac) was
swept up and then down while the oscillation amplitude in
the middle of the CNT was measured from the acquired
images in a SEM at room temperature and at a vacuum
pressureof∼10-6Torr.Toevaluatetheeffectofaddedmass
on the dynamic behavior of the nanoresonator, a small
amountofplatinumwasdepositedonthemiddleoftheCNT
with electron-beam-induced deposition, and its mass was
estimated from the measured dimension.
Figure 3B shows the acquired response spectrum for a
nonlinear nanoresonator incorporating a CNT of 2L) ∼6.2
µm and D ) ∼33 nm driven with an ac signal of 10 V
amplitude. The initiation of the oscillation started at around
4 MHz, near the natural resonance frequency of this doubly
clamped CNT. The amplitude of the resonance oscillation
increased continuously during the increasing frequency
sweep up to 14.95 MHz, at which point the amplitude
suddenly dropped to zero. This response closely resembled
what was modeled previously for an intrinsically nonlinear
nanoresonatorandcorrespondedtoaresonancebandwidth
of over 10 MHz. During the ensuing decreasing frequency
sweep, the resonator stayed mostly in a nonresonance state
until reaching the neighborhood of the natural resonance
R?) lim∆?f0
∆fdrop
∆?
)
fo
?rdrop(
rdrop
2rdrop
2- 1
2- 1)
(5)
FIGURE3.Fabricatednonlinearcarbonnanotubenanoresonatorand
its resonance response. (A) SEM images in the top view and tilted
view of a representative nanoresonator employing a CNT suspended
between and fixed at both ends on the fabricated platinum electrode
posts. The acquired response spectra of a CNT (2L ) ∼6.2 µm, D )
∼33 nm) nonlinear nanoresonator driven with ac voltage signals of
(B) 10 and (C) 5 V amplitude.
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frequenciesoftheCNT,wheretransitionsbacktoresonance
oscillations occurred. By fitting the obtained drop-jump and
up-jump frequencies with the model prediction, the driving
force was estimated to be ∼7 pN and the Q factor of the
system was ∼260, which were in agreement with the
estimate from an electrostatic analysis based on the experi-
mental setup (Supporting Information) and the reported
Q-factorvaluesfortypicalCNT-basedresonators,28respectively.
Theoccurrenceofmultipleup-jumptransitionsduringthe
decreasing frequency sweep appears to be due to the
existence of multiple natural resonance frequencies in a
multiwalled CNT and thus multiple modes of resonance. In
theory,29there are the same number of fundamental fre-
quencies and resonance modes as the number of cylinders
in a multiwalled CNT. In a recent computational study,30it
was shown that in the strongly nonlinear regime there can
be coupling between multiple radial and axial modes of a
double-walled CNT, with van der Waals forces provoking
dynamic transitions between the modes of the inner and
outer walls. Such strongly nonlinear modal interactions can
be studied using asymptotic techniques in the context of
coupled nonlinear oscillators.31
The existence of multiple natural modes in this multi-
walled CNT-based nonlinear resonator can also be revealed
in an increasing frequency sweep when the driving force is
reduced. Figure 3C shows the response spectrum acquired
fromthesameresonatorwhentheappliedacamplitudewas
reduced to 5 V. Two distinct resonance modes were excited
in this case. The first mode appeared at around 4 MHz, and
its drop jump occurred at 7.05 MHz. The second mode was
then initiated right after the drop jump of the first mode and
jumpeddownat14.15MHz.Asshownpreviously,whenthe
driving force was increased, it appeared that the first mode
resonance became dominant and suppressed the initiation
of the second mode in the increasing frequency sweep
whereas in the decreasing frequency sweep, because there
was no dominant mode, those modes were excited in the
neighborhoods of their linearized resonance frequencies.
Similar observations have been reported in coupled nonlin-
ear resonators20but not, until now, for a multiwalled CNT
intentionally operated in a highly nonlinear regime.
The mass-sensing capability of the nonlinear nanoreso-
nator is evaluated by adding a small platinum deposit to the
middle of a suspended CNT, as shown in Figure 4. In this
case, the CNT is ∼6.0 µm long and ∼26 nm in diameter.
Theaddedmasscausedbotha2.0MHzshiftofthelinearized
natural frequency, approximately defined as the frequency
where the resonance oscillation was initiated, and a more
significant 7.4 MHz shift of the drop frequency. The added
mass was estimated to be ∼7 fg on the basis of the dimen-
sions of the deposit measured from the acquired SEM
images (Supporting Information). The corresponding mass
responsivity calculated from the shift in the drop frequency
(Rm,nonlinear) 1.06 Hz/zg) was 3.7 times that calculated from
the linearized natural frequency (Rm,linear ) 0.29 Hz/zg).
These mass responsivity values compare favorably with our
model prediction from which Rm,nonlinear) 2.18 Hz/zg and
Rm,linear) 0.60 Hz/zg. The magnitude of the shift in the drop
frequency increases with the increase in added mass while
in the meantime the bandwidth of the resonance decreases
(Supporting Information).
This demonstration of a relatively simple nonlinear nan-
oresonator incorporating intrinsic geometric nonlinearity
offers a model system for expanded studies of the nonlinear
resonance behavior, which has been shown to be rich in
physicsandinopportunitiesforpracticalapplicationsonthe
macroscale, now down to the nanoscale. In this study, we
show a prototype as a mass sensor that can be applied to
other types of high-sensitivity sensing applications. Com-
paredwiththesensingprinciplesappliedinnanoscalelinear
resonance systems, a nonlinear resonance system can
exploit the instabilities intrinsically existing within the sys-
tem that are very sensitive to external perturbation. The
large oscillation amplitude and the sharp transition at these
bifurcation points are all very favorable characteristics from
the precision measurement point of view, which can poten-
tially enhance the measurement sensitivity of a practical
sensing system; the large oscillation amplitude implies less
susceptibility of the resonance system to thermal noise, and
a sharp transition allows for a narrow measurement band-
width.However,theuseofsuchanonlinearsystemforhigh-
sensitivity sensing applications may ultimately rely on our
more detailed understanding of the robustness of such
transitionsrelatedtotheinstabilityandtheeffectofexternal
noise (thermal noise and stochastic perturbations).21As
shown in other studies,19-21,23the complex dynamics of
FIGURE 4. Mass sensing with a nonlinear carbon nanotube nan-
oresonator. (A) SEM image showing the Pt deposit in the middle of
a suspended CNT (2L ) ∼6.0 µm, D ) ∼26 nm). (B) The acquired
response spectrum of this CNT nonlinear nanoresonator (O) before
and (•) after depositing a center mass with electron-beam-induced
deposition.
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nonlinear systems and the instabilities associated with it are
theoretically predictable and are robust enough for practical
use in sensing applications.
Moreover, nonlinear resonance systems have recently
been explored for more effective energy harvesting32and
more efficient mechanical damping applications because of
theirbroadbandresonancenatureanduniquecharacteristics
favoring directional energy transfer26in coupled systems.
As demonstrated in this study, such broadband resonance
behavior is preserved on the nanoscale and thus can be
potentially exploited for nanoscale energy-harvesting and
energy-transfer applications.
The design and demonstration of a simple nonlinear
mechanical resonator, which operates on the nanoscale,
expands the bandwidth of the resonance response, is tun-
able over a broad frequency range, and provides the inher-
ent instabilities that can be exploited for sensing applica-
tions, offers new conceptual strategies for the development
of nanoscale electromechanical devices. Such development
isfurtherfacilitatedbytheinherenteaseofrealizingintrinsic
geometric nonlinearity in a nanoscale resonator and can
thus be readily integrated into the ongoing development of
nanoscaleelectromechanicalsystemstoextendtheiroperation.
SupportingInformationAvailable.Derivationofthedrop
frequency. Estimation of the applied driving force. Young’s
modulus and the natural frequency of carbon nanotubes.
Added mass produced with electron-beam-induced Pt depo-
sition. Shift in the drop frequency and decrease in the
bandwidth with increasing center mass. This material is
available free of charge via the Internet at http://pubs.
acs.org.
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