Nonresonant high frequency excitation of mechanical vibrations in graphene based nanoresonator

We theoretically analyse the dynamics of a suspended graphene membrane which is in tunnel contact with grounded metallic electrodes and subjected to ac-electrostatic potential induced by a gate electrode. It is shown that for such system the retardation effects in the electronic subsystem generate an effective pumping for the relatively slow mechanical vibrations if the driving frequency exceeds the inverse charge relax- ation time. Under this condition there is a critical value of the driving voltage ampli- tude above which the pumping overcomes the intrinsic damping of the mechanical resonator leading to a mechanical instability. This nonresonant instability is saturated by nonlinear damping and the system exhibits self-sustained oscillations of relatively large amplitude.

material for the construction of nanoelectromechanical resonators. These graphene features are of great interest both for fundamental studies of mechanics at the nanoscale level and a variety of applications, including force, position and mass sensing. [1][2][3][4] In particular, it was demonstrated 5 that the graphene-based nanomechanical resonator can be employed as an active element for frequency-modulated signal generation and efficient audio signal transmission. Operation of most of the nanomechanical devices is based on the excitation of mechanical vibrations by an external periodic force, of electrostatic or optic origin, with a frequency comparable with the vibrational frequency of the mechanical resonator. At the same time, it was early shown 6,7 that in a certain nanoelectromechanical systems self-sustained mechanical oscillations with relatively large amplitude may also be actuated by using the phenomenon of "shuttle instability". In the shuttle structures described in, 6,7 the instability was found to occur at driving frequencies which are much smaller compared with the eigenfrequency of the mechanical subsystem. In the present work we are seeking to answer the question if it is possible to achieve a regime of selfsustained oscillations in a graphene-based nanoresonator by using an electromechanical instability effect caused by a nonresonant driving field. In the paper we demonstrate that such a possibility really exist. However, the electromechanical instability in the graphenebased resonators similar to those considered in the publication 1-5 occurs when the driving frequency is much greater than the eigenfrequency of the mechanical subsystem, in the contrast to the shuttle instability.
The system under consideration is shown in 1. A doped graphene membrane is suspended over a trench and separated from the grounded metallic substrate by an oxide layer. A top-gate subject to an AC voltage V G = V G cos(Ωt) induces an electrostatic potential on the graphene sheet ϕ(t). The potential drop between the membrane and the substrate creates an electronic flow through the tunnel barrier between them supplying electrons to the graphene sheet. At the same time, the high frequency electrical field E(t) between the gate electrode and the substrate will generate a time-varying force applied to Ṽ e e F Figure 1: A sketch of the graphene membrane resonator suspended over a trench and separated from the grounded metallic substrate by an oxide layer. A top-gate subjected to an AC voltage induces an electrostatic potential on the graphene sheet which depends on the membrane deflection.
the charged membrane. This coupling provides a feedback between the electronic and the mechanical subsystems. Below we show that due to the feedback, the electromechanical instability can occur leading to the high amplitude mechanical oscillations even for the gate voltage frequencies Ω much higher than the mechanical frequency ω m .
For the quantitative description of the above mentioned phenomena, we assume that the graphene sheet is an elastic membrane which motion is described within the continuum mechanics approach. Since the instability under investigation is due to a nonresonant phenomena we disregard the geometric nonlinearity of the graphene membrane. We assume that the dynamics of the membrane is completely characterized by the amplitude of its fundamental bending mode u(t), which time evolution we describe by a damped oscillator equationü with the electrostatic force where C S,(G) (u) is the mutual capacitance between the graphene and the substrate (gate) which depends on the amplitude of the fundamental mode. In the following consideration we study the case when ∂C G /∂u| 0 = −∂C S /∂u| 0 = β. This condition can be achieve by varying the distance between the top-gate and the membrane. Since the deflection amplitude is much less than the characteristic distance to the gate, one can find the power series expansion of the capacitance with respect to u and take C G (u = 0) + C S (u = 0) = C, Taking into account that the electrostatic potential ϕ(u) = (C G (u)V G + q(t))/C, one can obtain the following equation where q(t) is the accumulated charge on the graphene sheet and E 0 = βV G /C is the effective electrical field. The time evolution of q(t) is given by the second order equations of the corresponding driven RLC-circuiṫ where R is the tunnel resistance between the graphene and the substrate and L is an effective inductance of the RLC circuit. The set of (Eqs. 1, 3 and 4) describes the coupled dynamics of the electronic and mechanical subsystems. The solution of (Eq. 4) may be presented in the form where G(t) is the response function for the RLC-circuit which decays exponentially with Hence, the instant charge on the membrane is the time-delayed response to the external driving field with exponentially decaying memory. The decay time τ depends on the relation of two frequencies characterizing the RLC-circuit, ω RC = 1/RC and ω LC = 1/LC. In the overdamped limit, (ω RC /ω LC ) 2 = α ≪ 1, the memory function decays as exp(tω RC ) while under the underdamped condition, α ≫ 1, the decay function scales as exp(ω 2 LC t/2ω RC ) . We assume that the loss of memory is much faster than the timescale characterizing the dynamics of the mechanical system, ω m ≪ min{ω RC , ω LC /ω RC }. This allows us to expand u(t ′ ) = u(t) −u(t)(t − t ′ )... so the resulting charge on the membrane, and the force acting on the membrane, is proportional to u with a small correction proportional tou. Averaging (Eq. 1) over fast oscillations yields the effective equation for the low frequency mechanical motion This is the equation of a damped harmonic oscillator under constant forceF, with renor- whereΩ = Ω/ω RC , and is the normalized frequency shift. In the equation we also introduce the notion of the effective damping given by the following expressioñ where is the normalized additional damping which depends on the gate voltage driving frequency. The renormalization of the frequency is due to the main term proportional to u. For low driving frequencies, the charge oscillations are in phase with the oscillating electromagnetic field. In this case, the vibrational frequency is decreased, as seen on 2.
On the other hand, when the driving frequency exceeds ω LC , the charge oscillations are out of phase with the field and cause the vibrational frequency to increase. The correction term proportional tou causes a shift of the damping in the mechanical subsystem, shown on 3. When the external frequency increases, the damping shift decreases and becomes negative for driving frequencies exceeding the inverse response time For a strong enough electrical field, which will be estimated later, the damping shift cancels the intrinsic damping and the mechanical vibration becomes unstable in a frequency interval. This phenomena might be drastically enhanced by increasing the inductance which also leads to a more narrow frequency interval. The vibrational frequency of the membrane is renormalized due to the external field. The renormalization increases with field strength and gives softer frequencies for Ω < ω LC and stiffer frequencies for Ω > ω LC . An increased inductance also increases the delay time of the charge which shifts the renormalization peaks towards lower frequencies. Now we investigate the response to a stochastic electromagnetic field E(t) with power amplitude E 2 0 and power spectrum density S(ω), the condition for negative effective damping becomes under the assumption that the system is ergodic. The renormalization of the mechanical When α increases, two sharp peaks with positive and negative effective damping, respectively, build up. With only a slight increment in driving frequency the dynamics may leave the heavily damped region and enter the peak with large negative effective damping. The numerical simulation of (Eqs. 1 and 4) agrees excellently with the analytical solution and is here plotted for ω m /ω RC = 10 −2 .
frequency is changed in an analogous manner with the kernel η(Ω) replaced by ∆(Ω).
For white noise S(Ω) = 1 and the integral χ = 0, hence in order to get into the region of negative effective damping the low frequency noise needs to be filtered out.
Further, to reach the instability, the amplitude of the electrical field has to be large enough to overcome the intrinsic damping. In the case of monochromatic driving and negligible inductance, the optimal frequency to achieve maximal pumping is Ω = √ 3ω RC and χ = −1/8. A rough estimation of the amplitude needed for this situation is obtained by assuming mass m ≈ 10 −18 kg, vibrational frequency ω 0 ≈ 10 6 s, quality factor γ/ω 0 ≈ 10 −4 , electrode to membrane tunneling resistance R ≈ 10 7 Ω, capacitance of the membrane C ≈ 10 −16 F and circuit inductance L = 0 H, which gives For a top-gate to membrane distance d = 100 nm, we need a high frequency alternating voltage of the order of 10 mV. The required field strength can be lowered by increasing the size of the oscillator, the tunnel resistance R or the inductance L.
We now turn to the saturation mechanism of the instability investigated above. If the system is driven resonantly, mechanical damping or nonlinearity in the mechanical subsystem, which shifts the effective vibrational frequency out of resonance, will saturate the system at large amplitude oscillations. However, since the negative effective damping studied here does not originate from a resonant phenomena, the mechanism of saturation has to be of a different origin. One probable mechanism is nonlinear damping 8,9 e.g.
taken on the form γ n (u −ū) 2 /a 2 0u with strength γ n , arbitrary scaling a 0 and stationary position of the membraneū =F/(mω 2 m ). Adding this to (Eq. 6) saturates the amplitude of oscillations A at if the effective damping is negative and A = 0 elsewhere. Finally, it is worth noting that the energy absorption rateẆ of the driving field is increased by the value when the vibration is actuated. Here withẆ * = ΩCE 2 0 a 2 0 , E 2 * = mω m /(R 2 C 3 ) and quality factor Q = ω m /γ L . Therefore the measuring of the energy loss of the driving field gives possibility to indicate when the instability is reached and provide information about the nonlinear damping.
To conclude, we have found that mechanical vibrations of a graphene oscillator may be actuated by a nonresonant high frequency electromagnetic field. This is since the intrinsic damping of the oscillator is reduced when the frequency of the electromagnetic field exceeds the inverse response time of the charge oscillations in the graphene membrane.
If the field strength is strong enough to overcome the intrinsic damping, the mechanical vibrations become unstable and saturate due to nonlinear damping. The phenomena should be detectable with the available experimental techniques not only in graphene oscillators but also in other electromechanical oscillators due to the robustness of the described mechanism.