As the important electric loss of a graphene resonator, intrinsic current loss has received increasing attention, but the existing research is limited to qualitative analysis and approximate calculation. Based on the microscopic behavior of carriers, we have accurately established the calculation model of induced current loss, which is in good agreement with the existing experimental results. Compared with the previous models, the model in this work can not only fit the inverse V-shaped QVdc curve well but also be compatible with the V-shaped QVdc curve, which is beyond the reach of the previous models. In addition, the calculation results show that selecting the appropriate gate voltage combination when stimulating the graphene resonator can increase the quality factor by nearly 1–2 orders of magnitude. Furthermore, we reasonably explain the importance of DC gate voltage applied in the experiment compared with the case of non-DC gate voltage. This work realizes the accurate calculation of intrinsic current loss and is of great significance for reducing the intrinsic current loss in the practical application of graphene resonators.

The application of nano-resonators in the high-precision measurement is a frontier field of micronanoelectromechanical systems in recent years. Graphene is a single layer of carbon atoms tightly packed in a two-dimensional honeycomb lattice, which has excellent mechanical, electrical, and optical properties.1,2 In 2007, Bunch et al.3 first explored the potential of graphene as a resonator and promoted the follow-up research of related applications.4–6 

However, large loss and a low-quality factor are fatal shortcomings that hinder the practical application. The losses can result from various sources, such as air loss,7,8 attachment loss,9,10 thermoelastic loss,11,12 and loss due to defects,13 which are common for nano-resonators made of semiconductors. Since graphene is a conductor,14,15 the loss caused by the directional movement of carriers cannot be ignored.

Intrinsic current loss, which is one of the ohmic losses of graphene resonators, is the Joule heat produced by the periodic movement of surface carriers on the graphene film. In 2007, Seoanez et al.15 concluded that in contrast to semiconductor resonators, the loss of graphene resonators is mainly determined by the ohmic loss in the graphene film and metal gate after they studied various losses in graphene and nanotube resonators. However, the research on intrinsic current loss is rarely reported, and the existing research is confined on qualitative analysis and approximate calculation.15,16

In this work, we established an accurate calculation model for the intrinsic current loss of the graphene resonator based on the microscopic behavior of surface carriers during periodic vibration. Then, this model was compared with the previous calculation models and used to fit the experimental data. Finally, the relationship between the induced current loss and the gate voltage was analyzed. This work provides a theoretical basis to reduce the intrinsic current loss of graphene resonators at the same time.

The basic structure of a graphene resonator is shown in Fig. 1(a). Two sides of graphene are fixed on a Si/SiO2 substrate, and part of SiO2 is etched to obtain suspended graphene. Gold electrodes are plated on both ends of the graphene resonator. The gate voltage is applied between the gold electrode and the back grid of the substrate. At that time, graphene and the substrate can be equivalent to a parallel plate capacitor. The graphene film corresponds to one plate of the capacitor, and the silicon base is the other. It is a classical model for graphene resonators, whose feasibility has been fully verified in experiments.3,17,18

FIG. 1.

(a) Model of graphene resonators; the blue and red arrows represent the directions of current flow on the graphene surface during vibration. (b) Schematic diagram of the graphene resonator when at vibration. The gray dotted line represents the horizontal line, and the red dotted line represents the highest point and lowest point of the graphene film during vibration. The illustration shows the top view of the figure.

FIG. 1.

(a) Model of graphene resonators; the blue and red arrows represent the directions of current flow on the graphene surface during vibration. (b) Schematic diagram of the graphene resonator when at vibration. The gray dotted line represents the horizontal line, and the red dotted line represents the highest point and lowest point of the graphene film during vibration. The illustration shows the top view of the figure.

Close modal

The electrostatic excitation method with a better stability and a larger tuning range is adopted. The gate voltage is V=Vdc+Vaccosωt+ϕ0, where Vdc is the DC voltage, and Vac, ω, ϕ0 are the amplitude, the circular frequency, and the initial phase of the AC voltage, respectively. The electrostatic force of the graphene film is

F=ε0LW2d02[Vdc+Vaccos(ωt+ϕ0)]2.
(1)

With the condition of VdcVac, the electrostatic force is simplified as F=ε0LW(Vdc2+2VdcVaccos(ωt+ϕ0))/2d02. The first term is constant, which causes a stable lateral shift of the graphene film and increases the tension of the graphene film. The second term is the cosine excitation. When the driving frequency ω is equal to the natural frequency of the graphene film, the film resonates at ω. In the process of linear resonance, the distance between each point on the graphene film and the substrate is

d(x,y,t)=d0+d1(x,y)+Δd(x,y)cos(ωt+ϕ1),
(2)

where d0 is the etching depth of SiO2, d1(x, y) is the lateral displacement of each point caused by the constant electrostatic force and gravity, which can usually be omitted compared with d0, and Δdx,y is the vibration amplitude of each point. Figure 1(b) shows a schematic diagram of these parameters.

In the process of vibration, the instantaneous change of carriers at each point of graphene is

q(x,y,t)t=dV(t)dtΔC(x,y,t)+V(t)(ΔC(x,y,t))tA1(x,y)sin(ωt+ϕ0)+A2(x,y)sin(ωt+ϕ1)+A3(x,y)sin(2ωt+ϕ0+ϕ1)dxdy,
(3)
A1(x,y)=ε0ωd0Vac,A2(x,y)=ε0ωd02Δd(x,y)Vdc,A3(x,y)=ε0ωd02Δd(x,y)Vac,
(4)

where ΔCx,y,z=ε0dxdy/dx,y,z is the microelement capacitance, ɛ0 is the vacuum dielectric constant, and the simplification in Eq. (3) is based on the condition of Δd/d0 ≪ 1, where d0 is usually on the order of 100 nm,19–21 and Δd is on pm ∼ nm magnitude.3,22 The carriers produced during the vibration of graphene is equivalent to the charging and discharging processes of the capacitor. As the carriers increases, the current flows into the graphene film from the electrodes at both ends, and when the charge decreases, the current flows from graphene to both electrodes, as shown in Fig. 1(a). Due to the left and right symmetries, the current is zero at the center line of the graphene film. The current continuity equation is

q(x,y,t)t+j(x,y,t)x=0,
(5)

where j is the current density on the surface of graphene. Take the right half of graphene as an example. By integrating the differential equation in the direction of current flow, the current density at a certain point can be expressed as

j(xi,yi,t)=0xiq(xi,yi,t)tdx.
(6)

Combined with Joule’s law, the thermal power produced by intrinsic current on a microelement of graphene film is Pxi,yi,t=j2xi,yi,tRxi,yi, where R is the microelement resistance. The surface conductivity of graphene is σ=σmin2+σ2V, where σmin is the minimum surface conductivity at Vdc = 0,23 and its value is related to the quality and processing technology of graphene. σV=ε0VdcVμd0 is the conductivity offset induced by the bias voltage, where V′ is the Dirac point. So, the microelement resistance can be expressed as

R(xi,yi)=1σmin2+ε0VdcVμd02dxdy,
(7)

where μ is the graphene carrier mobility, and it can be considered constant when V0 is small. The thermal power of the entire graphene film can be obtained by integrating the space as Ptol = ∬ j2(x, y, t)R(x, y)dxdy. The total energy of graphene is expressed as the maximum kinetic energy E = 0.5 miω2∬Δd(x,y)2dxdy. Therefore, the expression of the corresponding quality factor is

Qq=2πEΔW=πmiω2Δd(x,y)2dxdy02πωj2(x,y,t)R(x,y)dxdydt.
(8)

In general, the vibration amplitude of each point of graphene corresponding to the first mode can be approximately expressed as16,24

Δd(x,y)=DcosπLx,
(9)

where D is the maximum amplitude of the center position. Thus,

Δd(x,y)2dxdy=12D2WL,
(10)
02πωj2(x,y,t)R(x,y)dxdydt=πωWL3ε02d02σVac212+D2Vdc22d02π2.
(11)

Substituting into Eq. (8), we can get

Qq=12miωd02D2σmin2+ε0VdcVμd02L2ε02112Vac2+D2Vdc22d02π2.
(12)

Since only the linear response of the resonance displacement to the electrostatic force and the surface current to the electric field is considered, this model of induced current loss is limited to be used in the linear range, and its extension to the nonlinear range is still to be studied. In addition, the circular fully clamped graphene resonator is considered, and the corresponding quality factor is calculated ( Appendix A).

In the previous research on induced current loss, the calculation of induced current was usually simplified. Sazonova et al.25 used the maximum current value as the periodic average value for calculation, resulting in larger results. Matthias Imboden26 and our previous studies16 considered the time variation of induced current, but ignored the spatial inhomogeneity of the induced current density on the surface of graphene. In addition, these three models do not take into account the first term of Eq. (3), ignoring the effect of periodic voltage. In the calculation with experimental data, we find this term is too large to be ignored. The expressions of these three models are shown in Table I.

TABLE I.

Summary of induced current loss models.

ReferencesModels
Sazonova et al.25  Qq=πkC(CVdc)21+(ωτ)2ωτ 
Imboden and Mohanty26  Qq=mωRVdc2C2 
Our previous work16  Qq=g0(1+βVdc)mωC2Vdc2 
This work Qq=12miωd02D2σmin2+ε0VdcVμd02L2ε02Vac212+D2Vdc22d02π2 
ReferencesModels
Sazonova et al.25  Qq=πkC(CVdc)21+(ωτ)2ωτ 
Imboden and Mohanty26  Qq=mωRVdc2C2 
Our previous work16  Qq=g0(1+βVdc)mωC2Vdc2 
This work Qq=12miωd02D2σmin2+ε0VdcVμd02L2ε02Vac212+D2Vdc22d02π2 

In our previous work,16 the variation of quality factor with the DC gate voltage is measured as shown in Fig. 2(a). The total loss is expressed as 1/Q(Vdc) = 1/Qm + 1/Qq(Vdc), where Qm is the quality factor when Vdc = 0, and Qq(Vdc) is the quality factor related to the induced current loss. The remaining models in Table I are used to refit the experimental data. It is worth noting that the formulas in Table I are relative to Qq, and so it is necessary to transform the fitting formula according to the above-mentioned relationship between Q and Qq. Since the fitting results of the first two models are consistent with our previous work, only the results in this work are shown as a comparison in Fig. 2(a). By fitting the measured results, we obtain an equivalent mass density of graphene as 2.5ρ0, which is smaller than the 9.8ρ0 in our previous work, where ρ0 = 7.6 × 10−19 kg/μm2 is the theoretical mass density of monolayer graphene.

FIG. 2.

(a) Model in this work used to refit our previous experimental data.16 (b) Equations (13) and (14) used to fit Chen’s experimental data. Source: Data extracted from Ref. 27.

FIG. 2.

(a) Model in this work used to refit our previous experimental data.16 (b) Equations (13) and (14) used to fit Chen’s experimental data. Source: Data extracted from Ref. 27.

Close modal

However, the gate modulation behaviors of the quality factor are complex and variable. The characteristic curve in Fig. 2(a) is an inverted V-shape, while, as shown in Fig. 2(b), it can also be a V-shape. For the V-shape curve, there is a significant difference in the fitting effect of different models. The first three models in Table I are similar, and the relationship between the quality factor, resonant frequency, and DC gate voltage can be unified as

Qq=ω(V)×(α+βV)γV2,
(13)

where α, β, γ are the fitting parameters, and in the first and second models, β = 0. The model proposed in this paper can be expressed as

Qq=ω(V)×a2+b2(VV)2c/V2+dV2,
(14)

where abc, and d are the fitting parameters.

Chen27 measured the relationship of graphene resonator between the quality factor and the gate voltage in the experiment. Since the original data and other graphene parameters cannot be obtained, we make a qualitative comparison. The measured resonant frequency ωV represented by the polynomial is substituted into Eqs. (13) and (14), and the two formulas are used to fit the quality factor in Fig. 2(b). Fitting with different Dirac point, an acceptable fitting can be obtained with V′ = −4 ∼ 0 V. The rationale for fitting parameters is discussed in  Appendix B. When the parameter c in Eq. (14) is large enough, c/V2 becomes the main term in the denominator. Thus, the quality factor no longer decreases monotonously but shows a decrease as Vdc increases. In the experiment, the resonance amplitude D is usually at the nanometer scale; d0 is on the order of 100 nm; and Vac is in millivolt level.28,29 Through careful calculation, it can be drawn out that, in Eq. (12), the order of Vac and DVdc/d0 is almost at the comparable level. That is, the role of parameter c in Eq. (14) cannot be ignored.

The electrostatic driving force of a unit area is

δF=ε0d02VdcVaccos(ωt+ϕ0)+ε04d02Vac2cos(2ωt+2ϕ0).
(15)

With VdcVac, the amplitude of the electrostatic driving force is

FAε0d02VdcVac.
(16)

The relationship between the resonant amplitude and the electrostatic driving force can be expressed as FADks, where ks is the stiffness of graphene. In the linear region, ks is kept constant, that is, the resonance amplitude is proportional to the driving force. We define a parameter A = Vac/D, then AVdcks. That is, when the Vdc is constant, A is proportional to ks. The quality factor is re-expressed as

Qq=12miωd02σmin2+ε0VdcVμd02L2ε02112A2+Vdc22d02π2.
(17)

On the basis of our experiments in the previous work,16 we further study the characteristics of the induced current loss varying with the gate voltage, keeping the driving force constant, and measured the resonance amplitude D = 40 pm. Here, we extend the resonance amplitude to 1 pm–1 nm. Since FADks, the curves of various D actually represent different values of ks. Therefore, we study the induced current loss performance of systems with different ks, since ks is usually constant for a certain linear system, which depends on the test conditions and device parameters. The variation trend of Qq with DC gate voltage is obtained by Eq. (17) under a certain value of D, as shown in Fig. 3(a).

FIG. 3.

(a) Dependence of the quality factor on the DC gate voltage with various values of D. The gray solid circles are the peak points extracted from different curves and connected by point line. (b) Dependence of the quality factor on the electrostatic driving force. The dotted line is under DC gate voltage with Vac = 1 mV.

FIG. 3.

(a) Dependence of the quality factor on the DC gate voltage with various values of D. The gray solid circles are the peak points extracted from different curves and connected by point line. (b) Dependence of the quality factor on the electrostatic driving force. The dotted line is under DC gate voltage with Vac = 1 mV.

Close modal

The red cross in Fig. 3(a) is extracted from the experimental data corresponding to Fig. 2(a). By fitting the data in Fig. 2(a) with the formula 1/QVdc=1/Qm+1/QqVdc, Qm = 1434 can be obtained. The value of Qq can be extracted by a simple operation. It is worth noting that the quality factor Qq in Fig. 3(a) considers only the induced current loss, which is different from the measured quality factor Q. With the increasing D, the quality factor considerably rises and the peak points of the quality factor move toward a smaller Vdc. With D larger than 102 pm, the peak is invisible in the studied range; thus, the quality curve exhibits a continuous decline. Since AVdcks, the value of A varying with Vdc is also different in these curves. According to Eq. (17), as Vdc keeps increasing, the impact of A declines in the denominator, which causes the curves to gradually converge. Thus, in order to reduce the induced current loss, it is essential to select an appropriate DC gate voltage. According to the calculation result, the quality factor can get almost one order improvement at the peak point.

When the Vdc is small, Qq decreases rapidly with the decreasing Vdc, which is due to the rapid increase of A. Furthermore, in the case of VdcVac, the driving force is re-expressed as F=ε0LWVac2/4d02+ε0LWVac2cos(2ωt+2ϕ0)/4d02, and the frequency of Vac should be set to half of the resonant frequency. The amplitude of the driving force is re-expressed as FAε0Vac2/4d02. Thus, the calculation formula of the corresponding quality factor should be revised to

Qq=6ωσminmid02ε02L2A2.
(18)

Assuming Vdc = 0, the driving force is determined only by the AC gate voltage. In order to compare with the case of VdcVac, we make FA equal in these two cases and compare the relationship between the quality factor and the electrostatic force. According to our previous experimental data,16 when the driving voltage VdcVac ≤ 21(VmV), the vibration is linear, and the driving force is proportional to the resonant amplitude. Therefore, the electrostatic driving force is set in the linear region. As shown in Fig. 3(b), the quality factor of applying a certain DC gate voltage is much larger than that without the DC gate voltage with the same amplitude of the electrostatic driving force, which also reasonably explains the importance of the DC gate voltage applied in the experiment.

Based on the periodic flow of carriers on the surface of graphene during resonance, we establish a new model for calculating the induced current loss. Compared with the previous models, it accurately reflects the microscopic behavior of carriers in space and time and takes the periodic variation of the gate voltage into account. The model can be consistent with the existing measured data. In addition, the model can fit a V-shape and an inverted V-shape of QVdc curve, which indicates a much stronger adaptability of our model compared with previous models. Then, we study the variation of quality factor with the DC gate voltage using this model. The DC gate voltage can effectively affect the quality factor by nearly one order of magnitude, and the voltage corresponding to the largest quality factor can be found through the QVdc curve. Furthermore, without the DC gate voltage, the quality factor usually will be much smaller than that by applying a certain DC gate voltage, even in the same electrostatic driving force, which theoretically explains the necessity of applying the gate DC voltage to excite the resonator. It should be noted that the induced current loss is not the main loss since the quality can be up to 105 by applying a suitable gate voltage. There still exists more influential loss to be explored. However, when other losses are suppressed, we propose a method to improve the quality factor.

This work was financially supported by the National Key R&D Program of China (Grant Nos. 2018YFA0306900 and 2017YFA0403200), the Natural Science Foundation of Hunan Province (Grant Nos. 2020JJ3039 and 2020JJ4659), the National Natural Science Foundation of China (Grant Nos. 51705528 and 51675528), and research projects under Grant Nos. JCYJ20170817111857745 and JCYJ20180504165721952.

The authors declare no conflict of interest.

Y.X. and F.H. contributed equally to this work. Y.X. performed simulation; Y.L. and H.F. defined the methodology; S.-Q.Q. was involved in conceptualization; F.H. and S.-Q.Q. supervised the work; Y.X., F.H., and Y.L. originally drafted the article; and M.-J., Z.X.-F.S., and J.-X.Z. reviewed and edited the article. All authors have read and agreed to the published version of the manuscript.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

For the strip graphene film fixed on both sides, when the carriers on the graphene surface changes, the flow direction of the current is shown in Fig. 1(a). Similarly, for the circular graphene resonators whose edges are all fixed, when the surface carriers increase, the current flows from the edge to the center, and when the surface carriers decrease, the current flows from the center to the edge, as shown in Fig. 4. As the graphene film vibrates in the ground state, its vibration mode is expressed as

Δd(r,θ)=DJ0β1(0)rl,
(A1)

where β10 is the first zero-point of the 0-order Bessel function. When the circular film resonates in the ground state, the mode shapes are uniformly distributed in the angular direction, and so the circular film can be divided into ring microelements in the radial direction. When the circular film resonates in the ground state, the mode shapes are uniformly distributed in the angular direction, and so the circular film can be divided into ring microelements in the radial direction. Driven by the electrostatic force, as shown in Eq. (1), the instantaneous carrier change of graphene microelements is expressed as

q(r,t)t=dV(t)dtΔC(x,y,t)+V(t)(ΔC(r,t))tA1(r)sin(ωt+ϕ0)+A2(r)sin(ωt+ϕ1)+A3(r)sin(2ωt+ϕ0+ϕ1)2πrdr,
(A2)
A1(r)=ε0ωd0Vac,A2(r)=ε0ωd02Δd(r)Vdc,A3(r)=ε0ωd02Δd(r)Vac.
(A3)
FIG. 4.

Schematic diagram of the circular graphene resonator.

FIG. 4.

Schematic diagram of the circular graphene resonator.

Close modal

Due to the current continuity equation, the current density on the surface of the circular film can be obtained as

j(r,t)=0rq(r,t)tdr.
(A4)

Accordingly, the equivalent surface resistance of graphene microelements can be expressed as

R(r)=1σmin2+ε0V0Vμd02dr2πr.
(A5)

By substituting the above formula into Eq. (8), the expression of quality factor can be obtained as

Qq=2πEΔW=πmiω2Δd2(r)dr02πωj2(r)R(r)drdt=8a3miωD2Rd04[J0(aR)2+J1(aR)2]ε02[a3Vac2R3d02+8V02D2(aRJ0(aR)22J1(aR)J0(aR)+aRJ1(aR)2)]σmin2+ε0V0Vμd02,
(A6)

where a=β1(0)R.

In Eq. (14), the parameters a, b, c, and d are redundant, that is, multiplying each parameter by the same factor can keep the fitting result constant. Therefore, we can evaluate the rationality of the fitting parameters by the ratio a/b and c/d. ab=σminε0μ/d0, where σmin is the minimum conductivity of graphene, and on the order of tens of μS; ɛ0 = 8.85 × 10−12 F/m is the vacuum permittivity; μ is the carrier mobility of graphene, and its value is generally in the order of 104 ∼ 105 cm2/V s at low temperature; d0 is the distance between the graphene film and the substrate, and the measured value is 295 nm, and so it can be concluded that the value of a/b is on the order of 10−1 ∼ 10°. cd=112Vac2Vdc2D22d02π2, where Vac is the AC gate voltage, which ranges from tens to hundreds of mW. Vdc is the DC gate voltage, which is measured in the range of −5 − 4 V. D is the resonant amplitude of graphene, which is generally on the order of 10−10 ∼ 10−9m. Therefore, it can be obtained that c/d is reasonable on the order of 102 ∼ 106. The fitting parameters of the five curves fitted by Eq. (14) are shown in Table II. It can be seen that the fitting parameters of the five curves are all in a reasonable range.

TABLE II.

The value of fitting parameters with Eq. (14).

Va/bc/d
0.40 2.54 × 103 
−1 0.32 3.74 × 103 
−2 0.11 2.13 × 103 
−3 0.71 1.67 × 103 
−4 0.96 1.28 × 103 
Va/bc/d
0.40 2.54 × 103 
−1 0.32 3.74 × 103 
−2 0.11 2.13 × 103 
−3 0.71 1.67 × 103 
−4 0.96 1.28 × 103 
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