Nonlinear resonances of a single-wall carbon nanotube cantilever

Nonlinear resonances of a single-wall carbon nanotube cantilever

I.K. Kim, S.I. Lee n

Department of Mechanical and Information Engineering, University of Seoul, Jeonnong-dong, Seoul 130-743, South Korea

H I G H L I G H T S

! The dynamics of an electrostatically actuated, cantilevered carbon nanotube (CNT) resonator are discussed by static and nonlinear dynamic analysis.! The CNT resonator exhibits linear and nonlinear primary, superharmonic, and subharmonic resonances depending on the DC and AC excitationvoltages.

! High electrostatic excitation leads to complex nonlinear responses such as a softening response branch and multiple stability changes at saddle-nodesor period-doubling bifurcation points near primary and secondary .

a r t i c l e i n f o

Article history:Received 14 August 2014Received in revised form28 November 2014Accepted 29 November 2014Available online 2 December 2014

Keywords:Carbon nanotubeNonlinear dynamicsBifurcationNonlinear resonance

a b s t r a c t

The dynamics of an electrostatically actuated carbon nanotube (CNT) cantilever are discussed by theo-retical and numerical approaches. Electrostatic and intermolecular forces between the single-walled CNTand a graphene electrode are considered. The CNT cantilever is analyzed by the Euler–Bernoulli beamtheory, including its geometric and inertial nonlinearities, and a one-mode projection based on theGalerkin approximation and numerical integration. Static pull-in and pull-out behaviors are adequatelyrepresented by an asymmetric two-well potential with the total potential energy consisting of the CNTelastic energy, electrostatic energy, and the Lennard-Jones potential energy. Nonlinear dynamics of thecantilever are simulated under DC and AC voltage excitations and examined in the frequency and timedomains. Under AC-only excitation, a superharmonic resonance of order 2 occurs near half of the primaryfrequency. Under both DC and AC loads, the cantilever exhibits linear and nonlinear primary and sec-ondary resonances depending on the strength of the excitation voltages. In addition, the cantilever hasdynamic instabilities such as periodic or chaotic tapping motions, with a variation of excitation frequencyat the resonance branches. High electrostatic excitation leads to complex nonlinear responses such assoftening, multiple stability changes at saddle nodes, or period-doubling bifurcation points in the pri-mary and secondary resonance branches.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Carbon nanotubes (CNTs) have excellent properties for use innanoelectromechanical systems. Since their discovery, [1] severalexperimental studies have been conducted on the electro-mechanical responses of CNT nanodevices under electrostaticfields [2–6], and many theoretical studies and simulations of theexperimental results have been performed. At the nanoscale,electrostatic [7] and intermolecular forces [8] are important fac-tors in the performance of nanoelectromechanical devices such asCNT-based tweezers [3,4,9,10], switches [7,11–18], memories[19,20], resonators [21–23], and bio- or mass sensors [24–29]. Inelectrostatically actuated nanodevices, it is important to

understand the effects of electrostatic and intermolecular forces asa function of excitation voltages. Rueckes et al. [19] used an en-ergy-based method to examine the bistability of electrostaticallyswitchable nanodevices for nonvolatile random access memory.Further, Dequesnes et al. [7,11,12] experimentally demonstratedthe feasibility of a nanoswitch [4] and predicted its static pull-involtage using linear [7] and nonlinear beam theories and mole-cular dynamics [11,12]. Ke et al. [13–16] proposed an “on/off” na-noswitch model by calculating static pull-in and pull-out voltages[13,14], and they considered the effects of electrostatic chargedistribution [15] and concentration [16] on CNTs. Raseckh andKhadem [17] studied pull-in behaviors in the nonlinearities ofnanocantilevers as a function of the ratio of the beam length to thebeam/electrode gap. They found that the geometric nonlinearity ofCNTs causes stiffening, whereas their inertial nonlinearity leads tosoftening of the cantilevered beam. They also studied the pull-in

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Physica E

http://dx.doi.org/10.1016/j.physe.2014.11.0221386-9477/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail address: [email protected] (S.I. Lee).

Physica E 67 (2015) 159–167

instability of a bridged nanoswitch with thermal expansion ofCNTs under electrostatic forces [18].

The nonlinear dynamics of an electrostatically actuated CNTcantilever has been investigated theoretically. Several studies havebeen conducted on nanoresonators [21–24] and mass- or bio-sensors [25–30] that use linear or nonlinear responses of a re-sonating CNT cantilever. Ke [21] performed numerical simulationsand analytical predictions to study the time responses of a double-sided, actuated CNT resonator. Ouakad and Younis [22] revealedcomplex nonlinear dynamics of CNT resonators by examining thegeometric nonlinearity of a CNT and van der Waals forces. Theystudied nonlinear processes such as resonator softening and the“escape band” in the frequency domain under low and high DCand AC voltage loads. Kim and Lee [23] simulated the nonlinearresponses of a CNT cantilever by considering its geometric andinertial nonlinearities, as well as its attractive and repulsive in-termolecular forces. Caruntu and Luo [24] investigated the effectsof damping, excitation voltages, and van der Waals forces on a CNTcantilever under soft AC electrostatic forces. The frequency re-sponses obtained by the method of multiple scales and the five-term reduced order model with the linear beam theory werecompared. Many approaches have been proposed for detecting thefrequency shift in resonating CNT-based sensors due to mass at-tachment [25–30], including nonlinear phenomena such as stabi-lity changes and stiffening or softening effects [29,30]. For ex-ample, Kim and Lee [29,30] examined the tip mass effect on re-sonating CNT cantilevers by means of nonlinear dynamics, in-cluding geometric and inertial nonlinearities. According to pre-vious works [17,30], the geometric nonlinearity of CNTs causesstiffening, but their inertial nonlinearity leads to softening. How-ever, Kim and Lee [30] showed that when the tip mass increases,the geometric nonlinearity undergoes considerable changeswhereas the inertial nonlinearity undergoes only weak changes.Because various changes in nonlinearities can have a significantimpact on the performance of nanoresonators or nanoswitches,the geometric and inertial nonlinearities should be not ignored[17,29–32].

In the present study, the dynamics of an electrostatically ac-tuated CNT cantilever on a fixed graphene electrode are in-vestigated by theoretical static and dynamic analyses. Previousstudies [11,22] employed numerical techniques based on thecontinuum model for predicting static pull-in and pull-out vol-tages. However, in the present study, static pull-in and pull-outphenomena are studied by the minimum potential energy method[19] under variable DC voltages with an asymmetric two-wellpotential problem. This is because the profiles of the total potentialenergy, which consists of the elastic energy of the single-walledCNT (SWCNT), electrostatic energy, and the Lennard-Jones poten-tial energy, differ according to varying applied voltages. The non-linear dynamics of the CNT cantilever are simulated by numericaltechniques for various DC and AC excitations and are comparedwith the results of a previous study [22]. This previous study didnot consider the effects of inertial nonlinearity and did not discussthe superharmonic resonances on the resonating CNT cantilever.According to linear and nonlinear beam theories, CNT devices havedifferent responses to applied voltages [17,30,32]. Because thenonlinearities of CNTs can affect the pull-in behaviors [17] or dy-namic stabilities [30], the CNT cantilever, including its geometricand inertial nonlinearities, is modeled in the present study by theEuler–Bernoulli beam theory. In a previous study, we discussed theeffects of both these nonlinearities on a resonating CNT with tipmass variations [30]. However, the present study is focused onnonlinear resonance phenomena (i.e., a series of bifurcations, andprimary, superharmonic, and subharmonic resonances) of a CNT-based resonator owing to its geometric and inertial nonlinearities(the ratio of the CNT length to the CNT/electrode gap) and DC/AC

voltage variations. Effects of attractive and repulsive inter-molecular forces determined from the Lennard-Jones potentialmodel [8] are also considered. Previous works have reported thatthe intermolecular forces affect the pull-in instability and non-linear frequency response in a CNT-based model [7,10,22,24,32]. Inthe present study, we use the Galerkin approximation with asingle mode and numerical integration techniques. The reduced-order model helps study strongly nonlinear systems and largedeflection of a CNT. Nonlinear dynamics are analyzed by detectingglobal bifurcations in the frequency domain by using AUTO-07Psoftware [33] and their time responses. To aid the implementationof CNT devices, we predict highly complex nonlinear responsesnear resonances (primary, superharmonic (orders 2 and 3), andsubharmonic (order 1/2) resonances) and discuss static and dy-namic stabilities.

2. Modeling

Fig. 1 schematically depicts an electrostatically operatedSWCNT cantilever (from Refs. [22,23]). It is employed as a na-noswitch or a nanoresonator by controlling DC and AC voltages.We use the continuum model for simulation of the nanodevicesowing to the good agreement of its results [7,22] with those of anexperimental study [4]. The static responses of the CNT device arepredicted using the minimum potential energy method [19]. Thetotal potential energy between the CNT and the graphene elec-trode is the sum of the elastic energy Eelas, external potential en-ergy such as the electrostatic energy Eelec [7], and the inter-molecular energy ELJ [7,8] described by the Lennard-Jones poten-tial [8]:

E E E E . (1)total elas elec LJ= − −

Because the total potential energy varies with the excitationvoltage V and the gap D between the CNT and the electrode, Eq. (1)can be rewritten as

E k D D E D V x E D x12

( ) ( , )d ( )d , (2a)L L

total eq init2

0elec

0LJ∫ ∫= − − ˜ − ˜

⎡⎣⎢

⎤⎦⎥( )

E D V V( , )

2 log 1 1 1

,

(2b)DR

DR

elec0

2

2πε˜ =+ + + −

E DC R D R D DR R

D D R( )

( )(2 4 5 )2[ ( 2 )]

,(2c)

LJattraction 6

2 2 2 2

7/2

π σ˜ = + + ++

E D

C a a a a aR a

( )

(315 3360 6048 2304 128 )320 ( 1)

.(2d)

LJrepulsion

122 2 3 5 7 8

9 2 19/2

π σ

˜

= − + + + +−

In Eq. (2), keq is the equivalent stiffness of the CNT; Dinit(390 nm) and D are the initial and instantaneous gaps, respec-tively; R is the CNT radius (5.45 nm); L is its length (2500 nm); 0ε isthe vacuum permittivity ( J m8.854 pC2 1 1− − ); and a is D/Rþ1. In the

Fig. 1. Electrostatically actuated CNT cantilever.

I.K. Kim, S.I. Lee / Physica E 67 (2015) 159–167160

present study, the dimensions of the nanodevice based on theexperimental study [4] are discussed [7,13] and used to simulatethe static and dynamic responses [7,13,22,23]. Owing to the con-centration of electrostatic charge [16] at the free end of the CNT,with increasing excitation voltages, the cantilever is deflected as aquadratic function shape toward the electrode and the equivalentstiffness keq can be assumed as being EI L60 /(13 )3 [34]. The elec-trostatic and intermolecular energies per unit length

(E E E, , andelec LJattraction

LJrepulsion˜ ˜ ˜ in Eq. (2)(b,c, and d), respectively)

between the SWCNT and graphene have been reported previously[7,8,11,12]. The constants C6 and C12 represent the carbon–carbonattractive and repulsive interactions, with values of 15.2 eV Å6 and2.52 KeV Å12, respectively. The surface density σ of graphite is38 nm 2− .

Another approach to characterizing the dynamic responses of aCNT cantilever under electrostatic fields is via a differentialequation. From our previous studies [23,29,30], we obtained thenonlinear equation of motion of a CNT device considering theelectrostatic force qelec and the Lennard-Jones attractive and re-pulsive intermolecular forces qLJ [7,8]. The nonlinear beam equa-tion and boundary conditions (Eq. (3)) for a bending deflectionw x t( , ) were obtained by Hamilton's principle, where the relationsbetween strain energy and kinetic energy are as reported pre-viously [30,35]:

⎡⎣⎢

⎤⎦⎥

Aw EIw cw EI w w w

A w w w w x x

q q

[ ( ) ] d d

, (3a)

L

x x

0

2

elec LJ

∫ ∫ρ

ρ

¨ + ″″ + + ′⋅ ′ ″ ′ ′

+ ′ ¨ ′⋅ ′ + ′ ′

= +

w t w t w L t w L t(0, ) (0, ) ( , ) ( , ) 0. (3b)= ′ = ″ = ″′ =

The CNT density ρ is m1350 kg/ 3, and its Young's modulus E is1 TPa. The variables A, I, and c denote the cross-sectional area,moment of inertia, and damping coefficient ( A Q/n nρ ω ), respec-tively; and nω is the nth natural bending frequency with a qualityfactor of Q n.

A nondimensionalized equation of motion was derived usingthe nondimensional variables in Eq. (4):

w wD

R

RD

D

D wD

w xxL

t

t

,

,

1 ,

,

,

,(4)

n

n

init

init

init

init

ω ΩΩω

=

=

= −

= −=

=

=

⁎ ⁎

⁎

⁎ ⁎

⁎

⁎

where EI A L/ (( ) 3.516 and )n n2

12ω β ρ β Ω= = is the excitation fre-

quency. We discretized the nondimensionalized partial differentialequation (PDE) using the Galerkin approximation with the firstbending mode of the Euler–Bernoulli beam theory. The approx-imate deflection w x t( , ) of the CNT cantilever is given as

w x t y t x( , ) ( ) ( ), (5)1 1ϕ=

where y t x( ) and ( )1 1ϕ are the temporal function and the first nor-mal bending mode, respectively. The eigenfunction x( )1ϕ [36] is

normalized to satisfy x x( ) d 10

11

2∫ ϕ = . After discretization of the

nonlinear equation of motion using the single-mode

approximation, the ordinary differential equation [23,29,30] isgiven as

y t cy t y t y t

y t y t y t y t

x q x t q x t q x t

x

( ) ( ) ( ) ( )

( ( ) ( ) ( ) ( ) )

( ) ( , ) ( , ) ( , )

d . (6)

1 1 12

1 G 13

I 12

1 1 12

0

1

1 elec LJattraction

LJrepulsion∫

ω ηη

ϕ

¨ + + ++ ¨ +

= + +

The dimensionless terms of the geometric and inertial non-linearities are (40.441) and (4.597 )G I 1η η γ , respectively. The di-mensionless damping coefficient c is equal to

L Q D L( ) / and ( / )12

1 1 init2β γ = . The superscript (n) of the non-

dimensionalized PDE Eq. (6) is omitted for convenience. The ex-ternal forces in the Galerkin integration are approximated in anexplicit form for y t( )1 only, by using curve fitting rather than theintegration of the implicit form, combining x( )1ϕ and y t( )1 with theright side of Eq. (6) [23,29,30]. The approximate external functionsare

x q x t q x t q x t

xd

y td

y t

dy t

( ) ( , ) ( , ) ( , ) d

(1/ ( )) (1/ ( ))

(1/ ( )),

(7a)

m n

n

0

1

1 elec LJattraction

LJrepulsion

22

max 13

3

max 1

44

max 16

∫ ϕ

γ ϕ γ ϕ

γϕ

+ +

= − + −

+− +

V LD EI

C LD EI

C LD EI

,2

,64

,(7b)2

02 4

init2 3

62 2 4

init5 4

122 2 4

init11γ πε γ σ π γ σ π= = = −

where d 0.0282 = , d 0.0073 = , d 0.000884 = , m 0.68= , and n 3.5= .The explicit function given by Eq. (7a) is a reasonable approxima-tion of the original integrated function on the right side of Eq. (6)and enables acquisition of rapid solutions [23,29,30].

3. Potential energy analysis

To predict the static behavior of the CNT cantilever, we calcu-lated the static pull-in and pull-out DC voltages by using the

0 200 400 600 800 1000

-140

-120

-100

-80

-60

-40

-20

0

20

Gap (nm)

Pote

ntia

l ene

rgy

(aJ)

Fig. 2. Potential energies and equilibrium positions at various DC voltages. Thefilled circles indicate the equilibrium positions between the CNT and the electrode,and the dotted circle indicates the occurrence of pull-in instability. (For inter-pretation of the references to color in this figure legend, the reader is referred tothe web version of this article.)

I.K. Kim, S.I. Lee / Physica E 67 (2015) 159–167 161

minimum potential energy method and Eq. (2) [19]. Fig. 2 showsthe potential energy profile and its equilibrium location undervarying DC voltages. The cantilever exhibits an asymmetric two-well potential behavior [19] superimposed by elastic, electrostatic,and Lennard-Jones potential wells. The red circles in Fig. 2 indicatethe equilibrium positions or local minimum energy points be-tween the CNT and the electrode at each DC voltage. When VDC¼0,a single minimum point exists at the initial gap of 390 nm. AtVDC¼1.11 V, two equilibrium points exist, at 363.9 nm and 9.48 nm.Because of the increase in the DC voltage from 0 to 1.11 V, the CNTis now located at 363.9 nm. With increasing voltage, the energybarrier for contact of the cantilever with the electrode decreasesgradually. When VDC¼2.315 V, the energy barrier vanishes and theequilibrium point jumps to 6.58 nm from 177.3 nm, as shown inFig. 3. At 6.58 nm, the tip of the cantilever is close to or touchingthe electrode. This phenomenon is termed pull-in, and the pull-inDC voltage can be calculated or measured experimentally[4,7,13,22]. The static pull-in voltage calculated using the mini-mum potential energy method [19] (VSPI¼2.31 V) correlates withvalues reported in previous work [4,7,13,22]. In the pull-in state,where VSPI42.31 V, only one equilibrium point exists at the elec-trode in the monostable region as described in Fig. 3. Pull-outrefers to the reverse phenomenon. When the applied DC voltagedecreases from the static pull-in voltage to 1.11 V, a set of twoequilibrium points exists in the bistable region, such as the upperand lower solid lines in Fig. 3 (or two minimum-potential-energypoints marked as red circles in Fig. 2). Because the cantilever ispulled in and there is an energy barrier for its pull-out from theelectrode, its end (tip) is on the electrode, i.e., along the lowersolid line in Fig. 3. At VSPIo1.11 V, the energy barrier disappearsagain, and the equilibrium point jumps to 363.9 nm from 9.48 nm.This phenomenon is termed pull-out. In summary, for our model,the calculated static DC voltages for pull-in and pull-out are 2.31 Vand 1.11 V, respectively.

4. Nonlinear dynamic analysis

In Section 3, we discussed the static responses of the CNTcantilever under DC voltages. In this section, we present the dy-namic analysis of the cantilever under DC and AC voltages, byusing the continuation and bifurcation software AUTO-07p, whichcan detect bifurcations globally. Nonlinear behavior of nanocanti-levers under harmonic excitation with DC and AC voltages has

0 0.5 1 1.5 2 2.50

50

100

150

200

250

300

350

400

Applied voltage (V)

Gap

(nm

)

Fig. 3. Static behavior with varying DC voltages. The solid and dotted lines indicatestable and unstable equilibrium positions, respectively.

Fig. 4. Superharmonic resonances of order 2: (a) frequency responses, (b) appliedvoltages and time responses for steady state, and (c) phase space for steady state(after time t 900= ; for V 0.3655 VAC = ) under AC-only voltage excitation with

0.4898Ω = and Q1¼150. In (a), SN () denotes a saddle-node bifurcation and thedotted line represents the unstable branch.

I.K. Kim, S.I. Lee / Physica E 67 (2015) 159–167162

been investigated previously [22,23,29,30]. Saddle-node (SN) andperiod-doubling (PD) bifurcations [37], as well as superharmonic(order 2), subharmonic (order 1/2), and fundamental resonances[38] have been investigated through numerical calculation. How-ever, in previous work [22], nonlinear dynamic analysis was per-formed on cantilever-type resonators for combined DC and ACvoltages. Therefore, in the present work, to understand or predictnonlinear phenomena, we conducted studies under AC-only ex-citation, extremely high DC and low AC excitations, and minimalDC and AC excitations.

4.1. Superharmonic resonances under AC harmonic excitations

As shown in Fig. 4, AC-only harmonic excitation leads to thesuperharmonic (order 2) resonance of the CNT cantilever at ap-proximately half of the fundamental frequency. This result re-sembles that reported in previous work [39] on nonlinear re-sonances of a cantilevered microbeam under harmonic AC ex-citation. Occurrence of the superharmonic resonance is attributedto the quadratic term V2 in the equation of electrostatic force [37].Under AC-only voltage excitation, V2 in the electrostatic force canbe expressed as

V V tV

tcos2

(1 cos 2 ). (8)2

AC2 2 AC

2Ω Ω= = +

Superharmonic resonance occurs because of the tcos 2Ω term.The positive or negative coefficient of the cubic term leads tostiffening or softening, respectively, of the cantilever [37]. With anincrease in the excitation voltage, the cubic term in the autono-mous version of Eq. (6) becomes negative and thus has a softeningeffect. It has been reported that the electrostatic force also leads tosoftening [22,23,29,30]. At V 0.35 VAC = , the CNT cantilever exhibitsa linear superharmonic resonance branch, as shown in Fig. 4(a).However, when V 0.3566 VAC = , SN bifurcations occur at the su-perharmonic resonance branch, along with a slight softening ow-ing to the increase in the electrostatic force. At higher voltage(same frequency), the cantilever has a softened motion (red line).The cantilever stability changes at the SN bifurcation point, as canbe seen in Fig. 4(a), where the solid and dotted lines representstable and unstable solution branches, respectively. Fig. 4(b) showsthe applied voltages and time responses that were simulated byMATLAB [40] using the 4th-order Runge–Kutta method in steadystate for an excitation frequency 0.4898Ω = (initial conditions:w w(0) (0) 0= = ). The frequencies of the superharmonic responsescorrespond to the primary fundamental; that is, they are twice theexcitation frequencies. Under AC-only excitation, the periodic re-sponse in steady state is period-1 motion (Fig. 4(c)).

4.2. Primary resonances under DC and low AC harmonic excitations

Under combinations of DC and harmonic AC loads, the CNTcantilever has various responses. When the AC voltage is less thanthe DC voltage ( )Vfor 0AC

2 ≈ , V2 can be expanded as

V V V tV

t

V V V t

2 cos2

(1 cos 2 )

2 cos . (9)

DC2

DC ACAC2

DC2

DC AC

Ω Ω

Ω

+ + +

≈ +

Therefore, the cantilever shows primary resonance under DCand low AC loads. If V 0.02 VAC = and V 0.5 VDC = , linear primaryresonance occurs with a phase difference of π , as depicted in Fig. 5.At V 1.0 VDC = , the SN bifurcations occur at a primary resonancewith the softening effect. With increasing DC voltage, a frequencyshift and increase in amplitude occur. Similar frequency shifts withincreasing DC voltage have been observed previously [41]. Underextremely low AC excitation combined with high DC voltages

0.9 0.95 1 1.050

0.5

1

Nor

mal

ized

am

plitu

de

0.9 0.95 1 1.05-3

-2

-1

0

Phas

e

Normalized excitation frequency (Ω)

SNSN

SN

SN

DC = 0.5 V

DC = 1.0 V

Fig. 5. Frequency and phase responses at primary resonances atV 0.5 and 1.0 VDC = with fixed V 0.02 VAC = and Q1¼150. SN () denotes a saddle-node bifurcation. The solid and dotted lines represent the stable and unstablebranches, respectively.

0 200 400 600 800 1000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Nondimensional time (t)

Non

dim

ensi

onal

def

lect

ion

(w)

980 985 990 995 1000-1

-0.5

0

0.5

1

w

t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1N

orm

aliz

ed a

mpl

itude

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-3

-2

-1

0

Phas

e

Normalized excitation frequency (Ω )

SNSN

SN

SN

PD

PD

Fig. 6. Primary resonances: (a) frequency response and (b) time response for allstates at V 2.0 VDC = , V 0.01 VAC = , 0.63Ω = , and Q1¼150. In Fig. 6(a), SN () de-notes a saddle-node bifurcation, PD () denotes a period-doubling bifurcation, andthe dotted line represents an unstable branch.

I.K. Kim, S.I. Lee / Physica E 67 (2015) 159–167 163

(assuming very small oscillations near the electrode), a strongsoftening effect is observed at the primary resonance branch, asshown in Fig. 6(a), with SN and PD bifurcations. The dynamic pull-in of the cantilever occurs through the PD bifurcations [22].Therefore, it touches the fixed electrode, as is the case with dy-namic pull-in; however, because repulsive Lennard-Jones inter-molecular forces exist, the cantilever has a stable or unstabletapping motion. Fig. 6(b) shows periodic tapping motion withV 2.0 VDC = , V 0.01 VAC = , and 0.63Ω = [zero initial conditions:w w(0) (0) 0= = ], because the end of the cantilever is extremelyclose to the electrode and the initial conditions do not hold in thePD bifurcation.

4.3. Primary and secondary resonances under DC and AC harmonicexcitations

If sufficient DC and AC voltages are applied at the same time,the CNT cantilever has complex nonlinear responses because ofthe parametric excitation [38] of the cos Ω and cos 2Ω terms in theelectrostatic force:

V V V V tV

t2 cos2

(1 cos 2 ). (10)2

DC2

DC ACAC2

Ω Ω= + + +

Under low DC voltage and sufficient AC voltage, the cantileverexhibits linear primary and superharmonic (order 2) resonances[23,29,30]. When the DC voltage is increased to 1.0 V andV 0.2 VAC = , with Q1¼150, the cantilever exhibits linear super-harmonic, nonlinear primary, and nonlinear subharmonic (order1/2) resonance branches (see Fig. 7). These harmonic resonancesare similar to those observed experimentally for microresonatorsunder electrostatic loads [37,42]. In the case of superharmonicresonance, the frequency of the time response at approximatelyhalf of the primary frequency is close to the primary resonancefrequency [29,30], and the phase varies slightly near half of theprimary frequency (Fig. 7). The time response at V 1.0 VDC = ,V 0.2 VAC = , 0.483Ω = , and Q1¼150 was simulated as shown inFig. 8(a). The oscillation amplitude in steady state (Fig. 8) corres-ponds to the amplitude of the superharmonic resonance (Fig. 7). Insteady state, the periodic response is period-2 motion (Fig. 8(b)).

In the case of primary resonance, the cantilever exhibits asoftening with SN bifurcation (Fig. 7), while its end touches theelectrode at the onset of escape through PD bifurcation [22]. The

cantilever has unstable motions such as chaotic tapping, pull-in, ordisorder in the escape band [22]. As shown in Fig. 9, when thecantilever is actuated by 0.99Ω = , V 1.0 VDC = , and V 0.2 VAC =outside the escape band, it exhibits a stable dynamic response insteady state. The oscillation amplitude in steady state (Fig. 9)correlates with the amplitude of the primary frequency resonanceat 0.99Ω = (Fig. 7). However, inside the escape band or at its onset( 0.924Ω = , V 1.0 VDC = , V 0.2 VAC = ), the cantilever's unstable re-sponse is chaotic tapping motion (Fig. 10).

Subharmonic (order 1/2) resonance branches corresponding tothe softening effect occur with PD bifurcations that are almosttwice the fundamental frequency (Fig. 7), because of the electro-static parametric excitation [38]. Additional PD bifurcation is ob-served on the subharmonic resonance branch at the SN bifurcationpoint. We find that for subharmonic responses, the frequency ofthe time response is close to the primary resonance frequency[29,30]. The oscillation amplitude in steady state correlates withthe amplitude of the subharmonic resonance (Fig. 7) [29,30].

Under high electrostatic excitation, the CNT cantilever exhibitshighly complex behaviors, as shown in Fig. 11. In this case, simu-lation is performed with V 1.5 VDC = , V 0.2 VAC = , and Q1¼150. Thecantilever undergoes multiple stability changes at the SN and PDbifurcation points on the nonlinear superharmonic (order 2),

0 0.5 1 1.5 2 2.50

0.5

1

Nor

mal

ized

am

plitu

de

0 0.5 1 1.5 2 2.5-3

-2

-1

0

Phas

e

Normalized excitation frequency (Ω)

Fig. 7. Frequency and phase responses for V 1.0 VDC = , V 0.2 VAC = , and Q1¼150. SN() denotes a saddle-node bifurcation; PD () denotes a period-doubling bifurca-tion; and the solid and dotted lines represent stable and unstable branches,respectively.

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Nondimensional velocity (dw/dt)

Non

dim

ensi

onal

def

lect

ion

(w)

0 200 400 600 800 1000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Nondimensional time (t)

Non

dim

ensi

onal

def

lect

ion

(w)

Fig. 8. Periodic responses at superharmonic resonance (order 2): (a) time responsefor all states and (b) phase space for steady state (after t ¼900) in the super-harmonic resonance branch ( 0.483Ω = ) at V 1.0 VDC = , V 0.2 VAC = , and Q1¼150.

I.K. Kim, S.I. Lee / Physica E 67 (2015) 159–167164

primary, and subharmonic (order 1/2) resonance branches. Addi-tional PD bifurcations are observed on the three resonance bran-ches; these bifurcations were combined at each upper SN bi-furcation point. The primary and superharmonic (order 2) re-sonance branches approach the SN bifurcation points. The escapeband at the primary resonance branch is wide because of the highelectrostatic excitation at V 1.5 VDC = . With increasing AC voltage,the escape band widens [22]. Thus, the escape band is created bylarge oscillations from high AC voltages or large static deflectionsfrom DC voltages. If the cantilever has many PD bifurcations be-cause of high electrostatic actuation, there is a possibility of itexhibiting complex or unstable motions. The superharmonic (or-der 3) resonance branch is observed near one-third of the primaryresonance frequency. At this resonance, the frequency of the timeresponse at approximately one-third of the primary frequency isclose to the primary resonance frequency, as shown in Fig. 12(a)(V 1.5 VDC = , V 0.2 VAC = , 0.303Ω = , Q1¼150, and zero initial con-ditions). The maximum oscillation amplitude in steady state(Fig. 12(a)) corresponds to the amplitude of the superharmonic(order 3) resonance (Fig. 11). The frequency of the time response atthe superharmonic (order 3) resonance frequency is triple theexcitation frequency (Fig. 12(a)). The periodic response in steadystate is period-3 motion (Fig. 12(b)).

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Nondimensional velocity (dw/dt)

Non

dim

ensi

onal

def

lect

ion

(w)

0 200 400 600 800 1000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Nondimensional time (t)

Non

dim

ensi

onal

def

lect

ion

(w)

Fig. 9. Periodic responses at primary resonance: (a) time response for all states and(b) phase space for steady state (after t ¼900) in the primary resonance branch( 0.99Ω = ) at V 1.0 VDC = , V 0.2 VAC = , and Q1¼150.

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

-0.8

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-0.4

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0

0.2

0.4

0.6

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1

Nondimensional velocity (dw/dt)

Non

dim

ensi

onal

def

lect

ion

(w)

0 200 400 600 800 1000-1

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0

0.2

0.4

0.6

0.8

1

Nondimensional time (t)

Non

dim

ensi

onal

def

lect

ion

(w)

Fig. 10. Chaotic tapping responses at primary resonance: (a) time response for allstates and (b) phase space for steady state (after t 900= ) in the escape band( 0.924Ω = ) at V 1.0 VDC = , V 0.2 VAC = , and Q1¼150.

0 0.5 1 1.5 2 2.50.2

0.4

0.6

0.8

1

Nor

mal

ized

am

plitu

de

0 0.5 1 1.5 2 2.5-3

-2

-1

0

Normalized excitation frequency (Ω)

Phas

e

Fig. 11. Frequency and phase responses at V 1.5 VDC = , V 0.2 VAC = , and Q1¼150. SN() denotes a saddle-node bifurcation; PD () denotes a period-doubling bifurca-tion; the solid and dotted lines represent stable and unstable branches, respec-tively; and EB denotes the escape band.

I.K. Kim, S.I. Lee / Physica E 67 (2015) 159–167 165

As discussed above, the electrostatic force and the inertialnonlinearity of a cantilever lead to softening of the system but itsgeometric nonlinearity causes stiffening effects. Our model is in-fluenced by strong electrostatic force due to the long cantileveredCNT and relatively small gap to the electrode. However, if the CNTbecomes shorter and the gap becomes larger, the effect of geo-metric nonlinearity will dominate the system in comparison to theeffect of the electrostatic forces. We simulated the frequency re-sponses of a comparison model (a shorter CNT with a larger gap)having different dimensions (L¼1000 nm, Dinit¼800 nm,d 0.052 = , d 0.0053 = , d 0.000654 = , m 0.5= , and n 3.5= ). Fig. 13shows the frequency and phase responses at V 1, 2, 3DC = , and5 V with a fixed V 0.4 VAC = and Q1¼150. At V 1 VDC = , a linearresponse is observed. At V 2 VDC = , SN bifurcations occur at theprimary resonance branch, followed by the occurrence of slightstiffening. At V 3 VDC = , the cantilever has three stable and twounstable solutions at the SN bifurcation points. Owing to the effectof geometric nonlinearity under large deflection, the system first

shows a stiffening response. As the tip end of the CNT comes closerto the electrode with increasing electrostatic forces, the systemshows a softening response. Above V 3 VDC = , the cantilever hasonly softening motion owing to the strong electrostatic forces.

5. Conclusions

In this paper, the static and dynamic behaviors of a carbonnanotube (CNT) cantilever under electrostatic fields were dis-cussed. The static behavior was understood in terms of an asym-metric two-well potential model. The static pull-in and pull-outvoltages were calculated via the bistability of the cantilever, whichdepended on the applied DC voltages. The cantilever dynamics

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

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0

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(w)

980 985 990 995 10000

0.5

1

1.5

2

Nondimensional time (t)

Appl

ied

volta

ge (V

)

980 985 990 995 1000-0.5

0

0.5

Nondimensional time (t)Non

dim

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onal

def

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ion

(w)

Fig. 12. Periodic responses at superharmonic resonance of order 3: (a) appliedvoltage and time response for steady state and (b) phase space for steady state(after t 900= ) in the superharmonic resonance branch ( 0.303Ω = ) at V 1.5 VDC = ,V 0.2 VAC = , and Q1¼150.

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.040

0.2

0.4

0.6

0.8

1N

orm

aliz

ed a

mpl

itude

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04-3

-2

-1

0

Phas

e

Normalized excitation frequency (Ω)

DC: 1 V DC: 2 V DC: 3 V DC: 5 V

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Nor

mal

ized

am

plitu

de

0 0.5 1 1.5-3

-2

-1

0

Phas

e

Normalized excitation frequency (Ω)

DC: 1 V DC: 2 V DC: 3 V DC: 5 V

Fig. 13. (a) Frequency and phase responses at primary resonances atV 1, 2, 3, and 5 VDC = with fixed V 0.4 VAC = and Q1¼150 (L¼1000 nm andDinit¼800 nm) and (b) its enlarged plot near primary resonance. SN () denotes asaddle-node bifurcation. The solid and dotted lines represent the stable and un-stable branches, respectively.

I.K. Kim, S.I. Lee / Physica E 67 (2015) 159–167166

were simulated under the following specific excitation conditions:superharmonic resonances under AC harmonic excitations; pri-mary resonances under DC and low AC harmonic excitations; andprimary, superharmonic, and subharmonic resonances under DCand AC harmonic excitations. The cantilever exhibited linear andnonlinear responses to combined DC and AC voltages. Electrostaticforces led to softening of the cantilever with saddle-node (SN)bifurcation at the primary and secondary resonance branches.Under AC-only excitation, the superharmonic (order 2) resonanceoccurred at nearly half of the fundamental frequency. The timeresponse at the superharmonic resonance frequency exhibitedperiod-1 motion. Under DC and harmonic AC loads, the cantileverexhibited nonlinear phenomena such as softening, with SN andperiod-doubling (PD) bifurcations on the superharmonic, primary,and subharmonic resonance branches with increasing excitationvoltages. At order-2 or order-3 superharmonic resonances, thetime response showed period-2 or period-3 motion, respectively.The fundamental frequency downshifted with increasing excita-tion voltages. In the “escape band,” the cantilever exhibited ex-tremely dynamic and unstable responses or chaotic motion be-cause of PD bifurcations. The escape band widened with increasingexcitation voltages. Furthermore, high electrostatic loads causedmultiple stability changes such as SN bifurcations and PD cascadesat the primary and secondary resonances. To aid the im-plementation of CNT devices, we predicted highly complex non-linear responses near resonances (primary, superharmonic (orders2 and 3), and subharmonic (order 1/2) resonances) and discussedstatic and dynamic stabilities. Under high voltage loads or in theescape band, the nanoresonator becomes unstable and its usageunder these conditions should be avoidable. In order to obtain aCNT-based resonator with stable responses, we need to considerthe structural nonlinearities in the CNT and the excitation voltages.

Acknowledgment

This work was supported by the National Research Foundationof Korea (NRF) grant funded by the Korean Government (MOE)(No. NRF-2010-0011761) and by the Human Resources Develop-ment Program (No. 20124010203260) of the Korea Institute ofEnergy Technology Evaluation and Planning (KETEP) grant fundedby the Korea Government Ministry of Trade, Industry and Energy.

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