Electromechanical Models of Micro and Nanoresonators

. This work is devoted to numerical and partly experimental investigation of the electromechanical models of micro and nanoresonators. The main purpose of this resonators is detected the adherence of micro or nanoparticles and measurement its mass. The periodical oscillations of mono and many layers nanocapacitors, from which nanoresonator situated in electric field is constructed, are studied. The purpose of this work is creating of electromechanical models of MEMS and NEMS the aim of which is to identify the changes introduced into dynamic systems due to adhesion of micro or nanoparticles. It’s significant to note that the time of overcharge of the nano-capacitor is much less, then the period of excited oscillations. This fact gives the possibility to apply asymptotic methods in numerical investigation. Physical experiments similar in model to the electromechanical nanoresonators were carried out. This work is an extended review article based on the results of previous our works.


Fig.1. Monolayer nanoresonator
The characteristic sizes of graphene resonator: the length of layer span -500÷1000 nm, the width of layer -10 ÷ 20 nm, the gap between graphene layer and conductive surface is also 10 ÷ 20 nm.
Let's consider the process of motion of this system taking into account the electrical forces. The different mechanical model of graphene layer may be consideredthe stretched string, the beam with flexural stiffness and simultaneously under the influence of axial tension, elastic membrane, thin stretched plate etc. These models may be both linear and nonlinear, including geometrical and physical nonlinearity. In common case the equations of motions may be written in a view ( ) +̈= ( , ). (1.1) Hereflexure of graphene layer, ( )differential operator of elasticity, − operator of inertia, ( , ) − the attracted force, induced by assistance of electrical field. By the long-term exposure of constant EMF, the system becomes in equilibrium position, which can be found from the system of equation (1.1) with absence of inertial forсe where 0 ( )linearized operator ( ), 2 − eigen frequency of free oscillations increasing in series.
Near of resonance only one component from series (1.4) makes sense to keep, which corresponds to resonance frequency . As usual it may be 1 = ( ) 1 (1.6) The using of Galerkin procedure to equation ( Equation (1.7) and (1.8) are described the motion of electromechanical system, which consist of oscillation system with one degree of freedom and electric circuit consist of in-series EMF, resistor and capacitor. Interaction between mechanical system and electric circuit determines that electric field in capacitor, which creates the mechanical force affected on the oscillated mass. In the same time displacement of the mass determines the change of capacity.
In case then EMF is a source with harmonic voltage ( ) = 0 and at small time of capacitor discharging in compare with period of voltage, i.e. at fulfillment of condition 0 ≪ 2 , we may neglect the first summand of equation (1.8), when charge of capacitor in time will change by form: The electromechanical model of resonator, which is analogy of nano-resonator is showed in Fig.2 where parameter 2 = 0 0 2 2 2 0 2 is a ratio of amplitudes of electric energy and kinetic energy of oscillations with frequency and amplitude equals to the gap of capacitor. Evidently, we consider only the case, then < 1, i.e. maximum of flexure deformation of the graphene layer don't exceed the initial value of the gap.
Equation (1.11) consists from two nonlinear components. Last of them connects with nonlinear influence of electrical force, acting at the presence of electric field and indefinitely increasing at approach of graphene layer flexure to the value of initial gap. Really this force plays the role of negative stiffness. First nonlinear component ( ) connects with nonlinear elastic of graphene layer at its transversal deformation. Graphene layer by the method of its production fix so that its edges don't move either transversal or longitudinal directions. The stiffness of graphene layer on the expansion so large, that equivalent Young's modulus for graphene in 5 times more then by the constructed steels. The longitudinal tension at flexure of graphene layer will be increase owing to its stretching in longitudinal direction that stipulate the addition for its transversal stiffness. This is result of many physical experiments. As the analog to the naonolayer, let's consider the static deformation of string previously stretched, the force of tension of which is 0 . String clamped so, that displacements of its edges in transversal and longitudinal directions don't possible, and in middle is loaded by transversal force (Fig.3). Substituting the last expression into conditions of equilibrium and using demonstrable geometrical correlation between flexure and angle , we obtain connection between vertical force and flexure = 4 ( 0 + ( √ 1 + (2 ) 2 − 1))/√1 + (2 ) 2 (1.13) The linearized approximation (1.13) gives the classical expression for transversal stiffness so called «inextensible» string in the same time more correctly believes that its name is infinitely stretched. The first approximation gives cubic relation 14) The influence of cubic term is more strong than initial tension 0 is smaller and larger than longitudinal stiffness of string * is great. Let's introduce dimensionless variable = and time = , where −eigen frequency of small oscillation of string. Taking into account (1.14) let us overwrite equation (1.11) in dimensionless form Static equilibrium positions at different sign of are shown in Fig.4 a b The stable oscillations at an excitation dimensionless frequency of the order of unity occur near stable equilibrium position as it can be seen in  for equation (1.15) numerical experiment with scanning of exciting frequency was fulfilled. At first the Cauchy task is solved for exciting frequency close on resonance frequency at zero initial data. Then after the steady-state was came the Cauchy task is solved again for nearest value of frequency, and initial data was taken from solution for previous value of frequency in time of installation of steady-state regime. This procedure returns at many values of frequency. Responsible resonance curves are showed in Fig.6, Fig.7.  Demonstrated here resonance curves have characteristic breaking conditioned either elasticity nonlinear characteristic (at small initial tension), or nonlinear negative electric stiffness (at large initial tension and large amplitude of oscillations). Resonance curves with characteristic, which has breaking, observe in real physical experiments, in particular these results demonstrate in article [7].
In laboratory of «Mechanical and processes of control» department was produced the demonstrated device, which enables to observe «breaking» resonance curves. As oscillations system a segment of steel wire clamped by edge was used with possible of previous tension. The oscillation of wire is excited by magnetic system, consisted from steel magnetic conductor (spring), permanent magnet and coil with alternating current. The coil feeds from generator of standard signals with possible adjustment of amplitude and frequency. The dependence of magnetic force, acted to wire, from values of wire flexure is analogy to the same dependence of electric force, acted in graphene. Namely this force rises monotonous at increasing of flexure. The equation, described the oscillation of wire in this experiment, is analogy to equation (1.11). In Fig.8 and Fig.9 the examples of oscillogram and resonance curves, obtained on the experimental device, are shown.  The direction of scanning of frequency at execution of experiment is shown in Fig.9. On the resonance curve characteristic breaking at scanning of exciting frequency top-down is visible.

Differential resonator
The scheme of differential resonator, consisted from two graphene layers, was proposed in works [8,9]. Scheme of this resonator is shown in Fig.10. Differential resonator consists from two parallel graphene layersbasic and additional. Additional layer is situated under basic layer and represents the plate conductive surface. The source of permanent EMF (basic) is aimed for creation of force interaction between basic and additional layers, the source EMF (additional) is proposed for assign the initial condition. The principal of work of differential resonator consists in next. The aim of one cycle of measuring is that voltage from additional source in a form of shot impulse excites combined free oscillations of graphene layers. Partial eigen oscillations of each layer (without connection) are very close and connection between them is soft in comparison the natural elastic of each layer. Thus, this system has two graphene layers closely approximated one from another and close to partial frequencies. Thus, free oscillations of this system will have the character of beating. The envelope of this process is determinate by detection. The characteristic envelope frequency is equal to the half of difference of partial frequencies and much less then partial frequency of every layer. At adherence of particle on top layer the partial frequency of this layer decreasing. And characteristic frequency also is changed, at that the small alteration of partial frequency gives the large alteration of envelope characteristic frequency. After some period, the free oscillations are damped and cycle of measuring may be repeated.
In laboratory «Mechanical and processes of control» department as well as at first case demonstrational device for observe the beating process was prepared. This device consists from two parallel steel cantilever beams.
The new scheme of graphene resonator, in which the oscillation is created only by parametric action, is proposed. The scheme of this resonator is shown in Fig.11. Parametrical resonator consists from one graphene layer, enclosed by two conductive surfaces. This combination is formed the two capacitors with one common armaturegraphene layer. The sources of AC EMF with synchronous frequency connect to each of capacitor. Macro-model analogy to parametrical resonator, which was showed in Fig.11, is proposes in Fig.12 The capacities of capacitors are changed at displacement of central mass (that analogy flexure of graphene layer) and have a view where the same designations, that in section 1 are input. The equation (3.1) with account relation between period of overcharging of capacitor circuit and period of external sources 0 ≪ 2 may be written in dimensionless form: The initial position of equilibrium of this resonator coincides with neutral unstrained position of graphene layer = 0. At absent of external voltages this position is stable. However, at existence of external voltage this position began unstable. The linearized equation (3.2) has a form: This is Mathieu equation, in which from parameter 2 both coefficient of pulsation and eigen frequency are dependent. The border of unstable field is shown in Fig.13. For unstable equilibrium position the diapason of frequency between left and right branches is defined, that this diapason becomes narrow then Q-factor of oscillation system is less. That parametrical resonance is different from usual. This fact may be used for increasing the accuracy of measuring the resonance frequency at law Q-factor. At parametric oscillations closed on equilibrium position the amplitude is limited by the nonlinear of the oscillations system. Amplitude of stationary regime may be found by harmonic balance method, i.e. to search solution of equation ( In this paragraph the equilibrium and oscillation nearly situated graphene layers with harmonic currents exciting parametrical resonance oscillations is considered. These parametrical oscillations are used for determine the mass of nanoparticle adherence. The task of this type is a prolongation of our works [7][8][9], in which the oscillations of the different types of nano-electromechanical system (NEMS) excite by alternative electric field, are considered. In these works the change of the spectrum of mechanical oscillations of different types NEMS, conditioned by adherence of nano-particle, is determined by the alteration of amplitude-frequency characteristic in a region of parametrical or forced resonance. In this paragraph parametrical oscillations excite by alternative current conducted through two closely spaced graphene layers. The determination of nanoparticle mass, stuck to one of parallel graphene layers, consists in determination of shifting the zone of parametrical resonance.
The main problem of creating a resonator, as detector of particle adherence, is an increasing the sensitivity to nano-mass of this nanoparticle. The explanation of advantage the using of the parametrical resonance at adherent of nanoparticle on graphene layer at law Q-factor (order~100) was done in previous paragraph.
The scheme of installation of graphene layers is analogy that it was fulfilled in differential resonator [9]. In difference from differential resonator, the graphene layers are used as conductors with alternating current. In result around conductive layers the alternating magnetic field is excited. In every layer the alternating in time magnetic force is arise and mechanical oscillations are excited. It's shown that this action may get excitation of parametrical resonance. As it known the width of zone of parametrical resonance is narrowed at decreasing of Q-factor of electromechanical system. This condition may be useful for increasing the accuracy of measurement the frequency of resonator at its law Q-factor.
The scheme of this resonator is shown in Fig.15 Resonator consists from two graphene layers situated on some distance one from another. At every layer the current is conducted as it shown in Fig.15. The forces-distributed interaction between conductors with currents 1 , 2 in case of its even directivity attraction is determined by relation where -radius-vector, connecting the points of deformable conductors, initially lie on common normally [10]. At first let's fulfill the solution of task of static deformation initially parallel conductors at action of distributed attractive electromagnetic force. The boundary problem of symmetric deformation relatively dimensionless cumulative displacement of strings has a form [10] In result we obtain relation Another way for obtaining this equation is using Galerkin method, supposed approximation = sin . The projection conditions have a form In result we obtain nonlinear transcendental equation for coefficient = From this equation we also can find the dependence = ( ), which comparison with analytic solution (4.5) is shown in Fig.16  Let's found the value of current, at which the static deformation by attraction of graphene layers can be estimate, used parameters of graphene nanoresonator: length = 1000 , width = 10 , thickness = 0.3 , resonance frequency ̅ = 30 and Q-factor~100. From formula for frequency of free oscillation we find the tension force = 2 ̅ 2 / 2 = 1,5 • 10 −12 Used the expression for parameter 2 ≈ 1 (Fig.16)   Keeping in expansion (4.16) only first form, we may find approximate alteration of first eigen frequency, corresponded to static attraction of strings: As the function ( , ) = 2 (1− 0 ( , )) 2 is positive on the segment [0, 1] on coordinate , so integrand also is positiveintegral nonnegative, and, accordingly, the frequency of free oscillations taking into account attraction decreases. This decreasing is droningly to come up to zero at value of parameter = * , responsible to point of branching of static boundary problem. Namely in this case variational equation for nonlinear boundary problem (4.13) has non-zero eigen-function, coincide with solution of boundary problem (4.15) at Λ = 0. The dependence of first eigenfrequency taking in account attraction at increasing of parameter , i.e. considering of increasing of current amplitude, is shown in Fig.17.

Fig.17.
Here Λ 1relative frequency, i.e. divided on the first eigen frequency of string without taking into account their attraction, which in dimensionless time is equal .
In case of small oscillations, we consider approximate solution of boundary problem (4.20) in form ̃= 1 ( ) ( ), where 1 (s)first mode taking into account attraction of strings. Projected equation on this form, we have differential equation in time: The dependence of the zone of parametrical resonance from in mentioned approximation is determined by equal to zero the determinant of system (4.25) (Fig.18). The main idea of determination of nanoparticle mass, adherence to graphene layer, consists in alteration of parametrical resonance zone taking into account the change of eigen frequency and mode of free flexure oscillations. For simplification in further we assume, that significant alteration has place only with first symmetrical form and, thereafter, with first eigen frequency, what may be calculated, for example, at adherence of nanoparticle near string middle. At adhesion of nanoparticle both coefficients ( ), ( ) in equations (4.22), (4.23) and eigen frequency Λ 1 are changed. The aim of analytical represent of approximated zone of parametrical resonance consists in estimation of conformity spectral properties of nano-string alteration at adherence of the nanoparticle, with alteration of excitement frequency (frequency of current).
Let's consider forced and parametrical nonlinear oscillations of string taking into account damping, described by equation (4.12) in the assumption of viscous damping for considerable nano-scales. We will look the solution in a form ̃= 1 ( ) ( ), taken only first mode of free oscillations, which was founded only keeping the first mode of free oscillation taking into account attraction of the strings. Projective equation on the first mode of free oscillations, considered nonlinear components, can be written in a view: φ( )(̈+ 2̇) + ( ( )Λ 1 2 + 2 ( )) 2 − Λ 1 2 ( ) = 2 ( ) cos(2 ) By the same meaning, the main interest represents the behavior of system at the frequencies near to main parametrical resonance and, consequently, at value of current frequency near to free oscillations frequency. Let us stay on the determination of amplitude-frequency characteristic of parametrical resonance, believed that ~Λ 1 .
Attempts to receive the stationary regimen in zone of parametrical resonance by harmonic balance method give the system algebraic nonlinear equations, the analytic solution of which looks impossible. Therefore, the numerical experiment with nonlinear equation (4.26) was fulfilled as near so inside zone of parametrical resonance. In time of realization of numerical experiment, the Cauchy task solved firstly nearly of right border of parametrical resonance zone at zero initial condition. At achievement of steady-state regime the Cauchy task solved again for slightly smaller value of frequency, but at initial condition, taken from solution in some moment of time. This procedure repeated at many values of frequencies until the left border of parametric zone is reached. In Fig. 20, 21, 22, 23     From these oscillograms it's seen, that out of zone of parametrical resonance amplitude of oscillation extremely small and oscillations have the second harmonic of excitement frequency. Inside of parametrical resonance zone the amplitude receives the value is order 0.7 of initial distance between graphene layers and consist only first harmonic of excitement frequency.
Thereafter results of numerical experiment the resonance curve was built, which was shown in Fig.24.  Fig.24 The resonance curve in a zone of parametric resonance In Fig.24 the points designate the values of steady-state amplitudes of oscillations at appropriated values of frequency. Arrowheads are shown the direction of frequency alteration at transition from one regime to another.
Obtained zone of parametrical resonance is the enough narrow. The resonance curve turns out nonsymmetrical relatively to medium-sized of frequency band (zone of resonance). Moreover, we can observe the breaking on the resonance curve at scanner of exciting frequency top-down. To the small (relatively with zone of resonance) alteration of frequency corresponds abrupt decreasing of amplitude of steady-state regimes.
Thus, it was proposed the new scheme of graphene nano-resonator. Investigation of possible regimes of this resonator work is shown that in this device excitement of parametric oscillation with large amplitude is possible. The distribution of amplitudes on frequency diapason has resonance character and resonance curve has characteristic breaking.
Obtained results may be useful at using of this construction resonator as detector of mass of nanoparticle adherent on the one of its layers. For determination of size of adherent particle, it may be used the value of frequency at ampitude breaking, which more accuracy to measuring then the frequency of maximum on the resonance curve.

Self-oscillation regime of nanoresonator
In overwhelming majority of experimental works on study the graphene resonator [1][2][3][4][5][6], so as in works, where the various mechanical models of nano-resonators are considered [7][8][9], it's proposed to determine the eigen frequency of resonator by the method of building the amplitude-frequency characteristic (resonance curve). It means, in essence, the next. On the resonator the forced periodical action with some frequency is realized. Future the amplitude of steady-state oscillations of resonator is measured. After that the frequency of action is changed and again the amplitude of steady-state oscillation is measured. This procedure is realized in frequency diapason inside of which hypothetically the eigen frequency of resonator is situated. After that the dependence of amplitude of steady-state oscillations from frequency of action is built. The frequency corresponded to maximum value of amplitude assumes as required eigen frequency. The preference of this method is simplicity of experiment fulfillment.
Let's note the disadvantage of this method. At-first, the fulfillment of experiment demands the significant expenditures of time. The measuring must be carried out for large values of frequencies.
At second, the determination of maximum values of amplitude may have the large inaccuracy. Namely nearly of resonance the amplitude deeply changes at small alteration of frequency, that make difficult the searching of this extremum. Using nonlinear resonance curves, which have abrupt breaking, significantly simplify the task of searching resonance frequency. At that we have another unpleasant factorthe frequency of breaking strongly depends from amplitude of action.
Oscillating system, in which self-oscillations possible hasn't these enumerated imperfections. Appearance of self-oscillations regime is possible at availability of positive feedback between oscillations system and the sourсe of excitement of oscillations.
The important degree of self-oscillations regime is self-regulation on the resonance frequency at slow (in comparison with period of oscillation) alteration of parameters of oscillations system. For example, at adherent of nanoparticle on the graphene layer, the mass, taken part in oscillation process, is changed that stipulate to alteration of self-oscillation frequency.
The proposed scheme of self-oscillation resonator represents in Fig.25 Fig .25 The scheme of self-oscillation resonator Self-oscillation resonator consists from amplifier, which must have positive feed-back connection. Feed-back circuit properly has graphene resonator and vibration transducer of graphene layer.
Output current of amplifier conducts by graphene layer. Itself graphene layer is situated in magnetic field. Induction vector is normal to current direction and direction of layer flexure. In result the magnetic force acts on graphene layer and excites its flexure.
Graphene layer and conductive surface under layer are like, which produces capacitor. To armature of capacitor the source of constant EMF 0 is applied. At alteration of flexure of graphene layer the capacity is changed, that excite overcharge of capacitor. The current of overcharge depends from velocity of flexure alteration. Voltage applied to resistor is proportionately to current of overcharge and input on entry of amplifier, thereby feedback of amplifier close. This feedback may be positive or negative, that is determined of magnetic field direction.
Let us use mechanical model of graphene layer at its transversal oscillation. This model represents itself electromechanical system, consisted from oscillation mechanical system with one degree of freedom and electrical circuit from in-series source EMF, resistor and capacitor, presented in Fig.26. Interference of mechanical system and electric circuit stipulates due to exciting of mechanical force acted on the oscillation mass and displacement of mass excites the change of capacitance of capacitor. In a case of current transmission on graphene layer and at the present of magnetic field the magnetic force will be done on the mass. The value of this force is proportional to value of current 2 , magnetic induction and length of conductor (graphene layer) . herecoefficient of increasing at small input signals, 0voltage of limited signals on the exit of amplifier. This characteristic allows to taking into account almost linear sector at small input signals, smooth nonlinear at large input signals and limitation of output signal under some level, depended from feeding of amplifier. The voltage 0 must to choose from next consideration. At appearance of self-oscillating regime, the amplitude of output signal from amplifier will be closed to 0 . In steady-state self-oscillation regime it's necessary that amplitude of graphene layer will be enough large, but not be exceed the value of initial gap. From this at first it is necessary to solve the task about resonance forced oscillations of graphene layer under action of magnetic interference and to choose the value of output current of amplifier so, that amplitude of graphene layer in resonance regime will be specified value in order 0,3 -0,5 from the value from initial gap. Without losing generality it's may to choose that 0 = 0 , and necessary of output current provide by the value of resistor .
Equations ( to overcharge by one period of graphene oscillations. At too small value of this multiplierthe voltage on the resistor will be also small. Estimation of this parameter also is order 0,01 ~ 0,1. At last in equations (5.10) we have more one dimensionless multipliercoefficients of increasing . Choosing of this coefficient gives the possibility of excitation of oscillations.
Excitation of self-oscillations is possible then natural damping may be compensated by arrival of energy from external source. In a first from equation (5.10) we have two components, connected with velocity ′ . First from itsviscous friction, corresponded energy absorption, and second is a force, creating by interference of current and magnetic field, corresponded for feeding of energy. At conform choosing of amplifier coefficients it's may to obtain the positive balance of energy and as result -excitation of oscillations.
The fulfilled numerical experiment with equations (5.10) is shown that at corresponded of choosing the amplifier coefficient and limited voltage 0 it is may to obtain excitation of oscillation with output on the steady-state regime. In Fig.27 one from oscillogram obtained by numerical integration of the system (5.10) is shown. whererelative increasing of effected mass. In Fig.28 the oscillograms of steady-state self-oscillation at various values of additional mass are shown. The increasing of the mass of adherent particle is extended the period of steady-state selfoscillation and, consequently, the frequency decrease. Separately let us consider the case of small time constant of time of capacitor charging. Then in second from equation (5.10) we may to neglect the speed of alteration the capacitor voltage and in an explicit form to obtain the voltage on the capacitor. Evidently, that this system has two equilibrium position stable (center) and unstable (saddle). Equation (5.13) has energy integral which allows to find , corresponded to homoclinic separatrix. For its determination it is necessary to substitute phase coordinate of critical point (saddle). After that it is possible to find intersection of separatrix with abscissa axe 0 , solved the equation Existence of limit cycle is possible only at fulfillment the condition > 2 , i.e. at condition that inputting energy from source at small amplitudes more, then loosing energy for the account damping, and at large amplitude inversely smaller. Estimation of its amplitude gives the possible to show that limit cycle belongs to the area, which lies inside separatrix, determined by relation (5.15). The stable of limit cycle is determined by the sign of its characteristic ratio [11] ℎ = 1 ∫ (− 0 2 + − 3 2 + ) .

Conclusions.
Obtained results may be useful at construction of the graphene nanoresonator as detector of mass of nanoparticle adherence on the one of its layers. At adherence of nanoparticles to graphene layer its spectral characteristics are changed. The using of forced resonance for determination the eigen frequency of nanolayer has some disadvantages. The determination of maximum value of amplitude may have a large inaccuracy and requires a dense set of excitation frequencies. At second maximum of amplitude-frequency characteristic is difficult to determine at law Q factor. In our work, we propose other ways to determine the changes in spectral characteristics which have greater accuracy. For determination of size of stuck particle, it may be used the value of amplitude breaking, which take place in parametric resonance. In addition, with a low Q-factor, the parametric resonance zone narrows, which also gives an advantage in determination the spectral characteristics. In differential resonator free oscillations have character of beating. The characteristic envelope frequency is equal to the half of difference of partial frequencies and much less then partial frequency of every layer. At adherence of particle on graphene layer the partial frequency of this layer decreasing and the small alteration of partial frequency gives the large alteration of envelope characteristic frequency. The important advantage of self-oscillations regime is self-regulation on the resonance frequency at slow (in comparison with period of oscillation) alteration of parameters of oscillations system. For example, at adherence of nanoparticle on the graphene layer, the mass, taken part in oscillation process, is changed that stipulate to alteration of self-oscillation frequency.

Data availability Statement
The dataset, which was used and analyzed during the current study, at introducing physical parameters are available from the corresponding author Dr. Dmitry Skubov on request.

Conflict of interest.
The authors declared that have no conflict of interest.