ISSN 10283358, Doklady Physics, 2014, Vol. 59, No. 6, pp. 254–258. © Pleiades Publishing, Ltd., 2014. Original Russian Text © N.F. Morozov, I.E. Berinskii, D.A. Indeitsev, O.V. Privalova, D.Yu. Skubov, L.V. Shtukin, 2014, published in Doklady Akademii Nauk, 2014, Vol. 456, No. 5, pp. 537–540.
PHYSICS
Oscillation Stop as a Way to Determine Spectral Characteristics of a Graphene Resonator Academician N. F. Morozova, I. E. Berinskiia, b, D. A. Indeitseva, b,
O. V. Privalovab, D. Yu. Skubova, b, and L. V. Shtukina, b Received February 12, 2014
Abstract—A nanoresonator based on a graphene layer is investigated as an electromechanical oscillatory sys tem. Mechanical oscillations are excited in it by a highfrequency alternating electric field. A nanoresonator is considered as a capacitor with kinematically varying capacity of the determined transverse deformation of the graphene layer as one of its plates. In the case of small ratios of energy accumulated in a capacitor to the amplitude of energy of mechanical oscillations and the time constant of the capacitor charge to the period of free oscillations, excitation of both common and parametric resonances is possible. It is shown that upon decreasing the external frequency lower than the halffrequency of free oscillations, the cessation of forced oscillations of the nanolayer is observed. This makes it possible to determine more reliably the variations in the intrinsic frequency of the nanoresonator upon deposition of a nanoparticle on it. DOI: 10.1134/S1028335814060068
To determine the spectral characteristics most exactly, the amplitude–frequency characteristic (AFC) with a clearly pronounced discrete spectrum is necessary. The width of a resonant peak depends on the figure of merit of the system, which is determined by its dissipative properties. The introduction of the inertial switching on into such a system varies its spec trum, which can be used to reveal switching on itself. The accuracy of determining the resonant frequency depends on the width of the AFC resonant peak (the accuracy is inversely proportional to the figure of merit). Therefore, on the one hand, it is necessary to increase the figure of merit for the most exact determi nation of the spectral characteristics, and on the other hand, to decrease it in order not to miss the resonance under study. In this work, we assume use of the nonlin earity of the AFC nanoresonator situated under the effect of the ac electric field with the purpose of more stable determination of its varying resonant properties. New nanomaterials fabricated recently and the technologies of their use promote the development of new, in principle, nanoelectromechanical systems, particularly nanoresonators. Reviews of the modern
a
Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol’shoi pr. 61, Vasil’evskii Ostrov, St. Petersburg, 199178 Russia b St. Petersburg State Polytechnical University, ul. Politechnicheskaya 29, St. Petersburg, 195251 Russia email:
[email protected]
state of such systems and their potential applications are given in [1, 2]. One of the most widespread appli cations of resonators is their use as mass detectors. These systems include a thin film, the crystal, or the cantilever, which oscillates with a frequency up to sev eral hundred megahertz. Due to the adherence of the particle (molecule, atom) to a flexible surface, its res onant frequency changes. This effect makes it possible to determine the mass of the adhered particle. A serious disadvantage in using the graphenebased resonator is its low figure of merit (of about 100) [2, 3]. The latter is caused by the “Joule” dissipation, i.e., by the transformation of electric energy into thermal energy because of the appearance of eddy currents in the graphene layer itself [4, 5]. In this article, we consider a new, in principle, pos sibility of using such a resonator, which makes it pos sible to increase the measurement accuracy of the eigenfrequency at a low figure of merit of the oscilla tory system. The resonator based on the graphene layer is considered as the electromechanical oscillatory sys tem, in which mechanical oscillations are excited by the ac electric field in the space between the graphene layer and the conducting surface. Such a system is a capacitor with the capacity depending on the trans verse deformation of the graphene layer. The electric field is induced by the external ac voltage source. In contrast with the use of the linear AFC, we propose to take into account the nonlinear effects that follow oscillations in the electric field. They lead to a “soft”
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AFC with the cessation of oscillation. The measure ment of the cessation frequency is possible with a higher accuracy compared with finding the amplitude maximum of the linear AFC. The graphene layer and the conducting surface are the plates of the capacitor, which is connected to the ac voltage source. The graphene layer bends under the effect of mutual resistance forces between the capaci tor plates. This leads to varying the capacitor capacity depending on deflection. In the first approximation, we assume that capacity depends on the deflection of graphene layer x(t) S C = εε 0 , d0 – x ( t )
(1)
where S is the plate area and d0 is the distance between the graphene layer and the conducting surface in the absence of the electric field. The motions of the electromechanical system are described by the set of mutually related equations 2
Q dC 1 U mx·· – 2 + c g x = 0, Q· + Q = sin ωt, (2) dx RC R 2C where m is the mass of the graphene layer, cg is its bend ing stiffness, Q is the capacitor charge, and R is the resistance of the electric circuit supplying the ac volt age Usinωt to the capacitor. The first equation of set (2) describes mechanical oscillations of the graphene layer (capacitor plates) under the effect of the electric force of mutual attraction. The second equation of set (2) is the equation of the voltage balance in the electric circuit. Let us introduce dimensionless variables τ = λt and c q = Q , where λ = g is the eigenfrequency of bend Q0 m ing oscillations of the graphene layer and Q0 = C0U0 is the starting constant charge of the capacitor appearing under the effect of the supplied dc voltage U0, ξ = x . d0 As a result, we derive a set of equations with the “main” parameters, where the point denotes the derivative by the new “dimensionless” time τ, Ω = ω : λ 2 ·· ξ – αq + ξ = 0, Here, parameters 2
C0 U0 α = , 2 2 2mλ d 0
q· + κ ( 1 – ξ )q = u˜ κ sin Ωτ. (3)
1 κ = , RC 0 λ
˜ . u˜ = U U0
(4)
The physical meaning of the parameters intro duced is as follows: α is the ratio of the potential (elec DOKLADY PHYSICS
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tric) energy of the capacitor with the initial accumu lated charge Q0 = C0U0 to the kinetic energy amplitude during harmonic oscillations of the plate having mass m with intrinsic frequency λ and amplitude d0, κ is the ratio of the period of free mechanical oscillations of the capacitor plate to the damping time, or decreasing the initial charge of the capacity connected to resis tance R, and u˜ = U is the scaled amplitude of the U0 external harmonic voltage. The amplitude of plate oscillations far from reso nance or at a low applied ac voltage is small (small u˜ ). In this case, we can consider that the capacity is inde pendent of the deflection and Eqs. (3) are linear. Forced oscillations of the graphene layer with fre quency 2Ω occur in this case. Let us consider the equilibrium of the graphene layer under the effect of a constant electric voltage (Ω = 0). It is evident that the undeformed state of the graphene layer is not the equilibrium position in the presence of the dc electric field. Two equilibrium posi tions can occur, notably, a stable one with a smaller deformation and an unstable one with a larger defor mation; or, depending on the applied voltage, the equilibrium position may be completely absent. The account of the influence of the graphene layer on the capacitor capacity leads to three main conclu sions. First, the interaction force is equivalent to the presence of the elastic basis with nonlinear negative elasticity. Second, the external field excites the forced oscillations with a doubled frequency relative to the external effect frequency (electric voltage). Third, excitation of parametric oscillations is possible. Resonant modes are possible near Ω ⯝ 0.5 (the coincidence of the common resonance with the sec ond parametric one) and at Ω ⯝ 1 (main parametric resonance). We will seek the approximated solution of nonlin ear set (3) under the assumption of smallness of α = ˜ ˜ and a large value of κ = κ . The solution of set (3) εα ε can be found by a method similar to the van der Pol method. We seek variable ξ in the form of a quasihar monic function with “slow” coefficients as and ac at harmonics and with obligatory holding of slowly vary ing constant component a0: ξ ( τ ) = a 0 ( ετ ) + a s ( ετ ) sin τ + a c ( ετ ) cos τ, · ξ ( τ ) = a s ( ετ ) cos τ – a c ( ετ ) sin τ.
(5)
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ξ 0.4
0.6
0.6
0.2 0.1 0
0.8
0.8
0.3
100
200
300
400
0.4
0.4
0.2
0.2
0
500
−0.1 −0.2
100
200 300 400
−0.2
−0.2
−0.4
−0.4
−0.6
100
200 300
400
500 τ
−0.6
−0.8
Ω = 0.38
0
500
Ω = 0.50
Ω = 0.51
Fig. 1. Numerical solution ξ(τ) at various excitation frequencies.
ξ 0.8
Travel
Charge
q 1.5
0.6
1.0
0.4 0.5
0.2 0 −0.2
50
100 150 200 250 τ
−0.4
0
50
100 150 200
−0.5
250 τ
−1.0
−0.6 −0.8
−1.5 Fig. 2. Numerical solution Ω = 0.5.
Using the projection method, we derive the set of integrodifferential equations for slow coefficients: 2π
ε ˜ q2 a 0 = α dτ, 2π
∫ 0
2π
ε ˜ q2 a· s = α cos τ dτ, π
∫ 0
2π
2π
(6)
˜ q 2 sin τ dτ. a· c = – εα π
∫
2 ε α 1 – cos τ ˜ ⎛ u˜⎞ a 0 = dτ, 2π ⎝ ρ⎠ 2 [ 1 – b sin ( τ + ϕ ) ] 2 0
∫
2π
0
We seek the quasisteady solution of the second equation of set (3) neglecting the first summand. Such neglect physically corresponds to a small temporal constant of the electric circuit. In this case, we have the following expression for the charge: u˜ sin Ωτ u˜ sin Ωτ q = = . 1–ξ ρ – a s sin τ – a c cos τ
Let us consider the mode close to common reso nance, i.e., Ω = 0.5 + εδ with a small detuning from the resonance. In this case, Eqs. (6) take the form
2 1 – cos τ ˜ ⎛ u˜⎞ a· s = εα 2 cos τ dτ, ⎝ ⎠ π ρ 2 [ 1 – b sin (τ + ϕ)] 0
∫
2π
2 1 – cos τ ˜ ⎛ u˜⎞ a· c = – εα sin τ dτ, π ⎝ ρ⎠ 2 [ 1 – b sin ( τ + ϕ ) ] 2 0
∫
2
(7)
(8)
2
as + ac a where ρ = 1 – a0, b = , and tan ϕ = c . as ρ DOKLADY PHYSICS
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Analytic expressions are found for the integrals in set (8): 2π
I0 =
1 – cos τ
dτ ∫ 2 [ 1 – b sin ( τ + ϕ ) ]
π = ( 1 – b sin ϕ ), 2 2 (1 – b ) 1 – b Ic =
∫ 0
1 – cos τ 2 cos τ dτ 2 [ 1 – b sin ( τ + ϕ ) ]
2 bπ 1 ⎞ 1 ⎛ 1 – = sin ϕ + π cos ϕ 2⎝ ⎠ 2 2 2 b 1–b (1 – b ) 1 – b 2 1 1 ⎞ 1 – 2 ⎛ 1 – + π sin ϕ, ⎝ ⎠ 2 2 2 b 1–b (1 – b ) 1 – b (9) 2π
Is =
ξ 0.7 0.6
2
0
2π
257
0.4 0.3 0.2 0.1 0 0.44
0.46
0.48
0.50
0.52
0.54
0.56
0.58 Ω
Fig. 3. Amplitude–frequency characteristic.
1 – cos τ
sin τ dτ ∫ 2 [ 1 – b sin ( τ + ϕ ) ] 2
0
bπ = cos ϕ 2 2 (1 – b ) 1 – b 1 1 1 + ⎛ – 2 – ⎞ π sin 2ϕ. ⎝ 2 2 2 2⎠ b 1 – b b 2(1 – b ) 1 – b Set of equations (9) in the timeindependent case, when the amplitude of oscillations b, phase ϕ, and a0 κ are constants, with selected parameters εα = 0.1, = 10, ε ˜ and u = 1, has two solutions with different constants corresponding to a0 = 0.0455 and a0 = 0.7677 and π identical to b = 0.7071 and ϕ = 1.5708 ⯝ . The first 2 solution corresponds to the stable timeindependent periodic mode, and the second solution corresponds to the timedependent one. The layer motion with a smaller constant component and the larger amplitude is stable in the case of sustained oscillations. Let us present the results of numerical computation of the Cauchy problem for set of Eqs. (3) with the abovementioned parameters. Oscillograms ξ(τ) are plotted for cases Ω close to 0.5 (Fig. 1). Figure 1 represents the numerical solution ξ(τ) at various excitation frequencies. Oscillation cessation is clearly seen with decreasing values of Ω below 0.5. The latter indicates the soft character of nonlinearity. Figure 2 shows the displacement and charge oscil lograms at Ω = 0.5. The carrying frequency of the dis placement oscillogram exceeds twofold the frequency of charge varying, which equals the frequency of the applied voltage. DOKLADY PHYSICS
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The oscillation amplitude and the constant com ponent in the timeindependent mode, which were found by numerical integration, almost coincide with the result of the analytical calculation of set of alge braic Eqs. (8), (9) for the timeindependent case (first of solutions). Figure 3 shows the AFC of the established plate oscillations at Ω close to 0.5. Oscillation cessation at the external action frequency, which decreases relative to this value, is shown. Only the main parametric resonance takes place at the external voltage frequency equal to the free oscil lation frequency (ω = λ or Ω = 1). In this case, the motion amplitude of the graphene layer is much smaller than in the case of the usual mechanical reso nance. Based on our results, we proposed a new electro mechanical model of the graphene nanoresonator allowing for variation in the capacitor capacity appearing upon deforming the graphene nanolayer (one of the plates). It is shown that both common res onance and parametric resonance appear in the sys tem with the excitation of oscillations by the harmonic external voltage. In the case of the main resonance, when the frequency of the external effect (applied electric voltage) is close to half of the intrinsic mechanical frequency, the system has a soft AFC. As the external frequency lowers below the halffre quency of free bending oscillations, oscillation cessa tion of forced oscillations is observed, which makes it possible to determine more reliably the resonator eigenfrequency with the deposition of a nanoparticle on it.
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Further investigations of the proposed model can be performed in several directions. For example, the nonlinearity of elastic properties of the graphene layer may turn out to be essential at large deflections, or it may turn out to be necessary to take into account the distribution nonuniformity of the electric field inside the capacitor. ACKNOWLEDGMENTS This study was supported by the Russian Founda tion for Basic Research, project no. 140100845, and by the Program of the President of the Russian Feder ation, project MK4873.2014.1.
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Translated by N. Korovin
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