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 *Q* − *V*_{dc} curve well but also be compatible with the V-shaped *Q* − *V*_{dc} 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.

## I. INTRODUCTION

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.

## II. INDUCED CURRENT LOSS MODEL ESTABLISHMENT

The basic structure of a graphene resonator is shown in Fig. 1(a). Two sides of graphene are fixed on a Si/SiO_{2} substrate, and part of SiO_{2} 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}

The electrostatic excitation method with a better stability and a larger tuning range is adopted. The gate voltage is $V=Vdc+Vac\u2061cos\omega t+\varphi 0$, where *V*_{dc} is the DC voltage, and *V*_{ac}, *ω*, *ϕ*_{0} are the amplitude, the circular frequency, and the initial phase of the AC voltage, respectively. The electrostatic force of the graphene film is

With the condition of *V*_{dc} ≫ *V*_{ac}, the electrostatic force is simplified as $F=\epsilon 0LW(Vdc2+2VdcVac\u2061cos(\omega t+\varphi 0))/2d02$. 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

where *d*_{0} is the etching depth of SiO_{2}, *d*_{1}(*x*, *y*) is the lateral displacement of each point caused by the constant electrostatic force and gravity, which can usually be omitted compared with *d*_{0}, and $\Delta dx,y$ 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

where $\Delta Cx,y,z=\epsilon 0dxdy/dx,y,z$ is the microelement capacitance, *ɛ*_{0} is the vacuum dielectric constant, and the simplification in Eq. (3) is based on the condition of Δ*d*/*d*_{0} ≪ 1, where *d*_{0} is usually on the order of 100 nm,^{19–21} and Δ*d* 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

where $j\u20d7$ 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

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 $\sigma \u2032=\sigma min2+\sigma 2V$, where *σ*_{min} is the minimum surface conductivity at *V*_{dc} = 0,^{23} and its value is related to the quality and processing technology of graphene. $\sigma V=\epsilon 0Vdc\u2212V\u2032\mu d0$ is the conductivity offset induced by the bias voltage, where *V*′ is the Dirac point. So, the microelement resistance can be expressed as

where *μ* is the graphene carrier mobility, and it can be considered constant when *V*_{0} is small. The thermal power of the entire graphene film can be obtained by integrating the space as *P*_{tol} = ∬ *j*^{2}(*x*, *y*, *t*)*R*(*x*, *y*)*dxdy*. The total energy of graphene is expressed as the maximum kinetic energy *E* = 0.5 *m*_{i}*ω*^{2}∬Δ*d*(*x*,*y*)^{2}*dxdy*. Therefore, the expression of the corresponding quality factor is

In general, the vibration amplitude of each point of graphene corresponding to the first mode can be approximately expressed as^{16,24}

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

Substituting into Eq. (8), we can get

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).

## III. RESULTS AND DISCUSSION

### A. Verification and comparison of models

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 Imboden^{26} and our previous studies^{16} 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.

References . | Models . |
---|---|

Sazonova et al.^{25} | $Qq=\pi kC(C\u2032Vdc)21+(\omega \tau )2\omega \tau $ |

Imboden and Mohanty^{26} | $Qq=m\omega RVdc2C\u20322$ |

Our previous work^{16} | $Qq=g0(1+\beta Vdc)m\omega C\u20322Vdc2$ |

This work | $Qq=12mi\omega d02D2\sigma min2+\epsilon 0Vdc\u2212V\u2032\mu d02L2\epsilon 02Vac212+D2Vdc22d02\pi 2$ |

References . | Models . |
---|---|

Sazonova et al.^{25} | $Qq=\pi kC(C\u2032Vdc)21+(\omega \tau )2\omega \tau $ |

Imboden and Mohanty^{26} | $Qq=m\omega RVdc2C\u20322$ |

Our previous work^{16} | $Qq=g0(1+\beta Vdc)m\omega C\u20322Vdc2$ |

This work | $Qq=12mi\omega d02D2\sigma min2+\epsilon 0Vdc\u2212V\u2032\mu d02L2\epsilon 02Vac212+D2Vdc22d02\pi 2$ |

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*(*V*_{dc}) = 1/*Q*_{m} + 1/*Q*_{q}(*V*_{dc}), where *Q*_{m} is the quality factor when *V*_{dc} = 0, and *Q*_{q}(*V*_{dc}) 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 *Q*_{q}, and so it is necessary to transform the fitting formula according to the above-mentioned relationship between *Q* and *Q*_{q}. 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 2.5*ρ*_{0}, which is smaller than the 9.8*ρ*_{0} in our previous work, where *ρ*_{0} = 7.6 × 10^{−19} kg/*μ*m^{2} is the theoretical mass density of monolayer graphene.

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

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

where *a*, *b*, *c*, and *d* are the fitting parameters.

Chen^{27} 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 $\omega V$ 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*/*V*^{2} becomes the main term in the denominator. Thus, the quality factor no longer decreases monotonously but shows a decrease as *V*_{dc} increases. In the experiment, the resonance amplitude *D* is usually at the nanometer scale; *d*_{0} is on the order of 100 nm; and *V*_{ac} is in millivolt level.^{28,29} Through careful calculation, it can be drawn out that, in Eq. (12), the order of *V*_{ac} and *DV*_{dc}/*d*_{0} is almost at the comparable level. That is, the role of parameter *c* in Eq. (14) cannot be ignored.

### B. Effect of gate voltage on induced current loss

The electrostatic driving force of a unit area is

With *V*_{dc} ≫ *V*_{ac}, the amplitude of the electrostatic driving force is

The relationship between the resonant amplitude and the electrostatic driving force can be expressed as $FAD\u221dks$, where *k*_{s} is the stiffness of graphene. In the linear region, *k*_{s} is kept constant, that is, the resonance amplitude is proportional to the driving force. We define a parameter *A* = *V*_{ac}/*D*, then *AV*_{dc} ∝ *k*_{s}. That is, when the *V*_{dc} is constant, *A* is proportional to *k*_{s}. The quality factor is re-expressed as

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 $FAD\u221dks$, the curves of various *D* actually represent different values of *k*_{s}. Therefore, we study the induced current loss performance of systems with different *k*_{s}, since *k*_{s} is usually constant for a certain linear system, which depends on the test conditions and device parameters. The variation trend of *Q*_{q} with DC gate voltage is obtained by Eq. (17) under a certain value of *D*, as shown in Fig. 3(a).

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 $1/QVdc=1/Qm+1/QqVdc$, *Q*_{m} = 1434 can be obtained. The value of *Q*_{q} can be extracted by a simple operation. It is worth noting that the quality factor *Q*_{q} 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 *V*_{dc}. With *D* larger than 10^{2} pm, the peak is invisible in the studied range; thus, the quality curve exhibits a continuous decline. Since *AV*_{dc} ∝ *k*_{s}, the value of *A* varying with *V*_{dc} is also different in these curves. According to Eq. (17), as *V*_{dc} 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 *V*_{dc} is small, *Q*_{q} decreases rapidly with the decreasing *V*_{dc}, which is due to the rapid increase of *A*. Furthermore, in the case of *V*_{dc} ≪ *V*_{ac}, the driving force is re-expressed as $F=\epsilon 0LWVac2/4d02+\epsilon 0LWVac2\u2061cos(2\omega t+2\varphi 0)/4d02$, and the frequency of *V*_{ac} should be set to half of the resonant frequency. The amplitude of the driving force is re-expressed as $FA\u2248\epsilon 0Vac2/4d02$. Thus, the calculation formula of the corresponding quality factor should be revised to

Assuming *V*_{dc} = 0, the driving force is determined only by the AC gate voltage. In order to compare with the case of *V*_{dc} ≪ *V*_{ac}, we make *F*_{A} 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 *V*_{dc} ⋅ *V*_{ac} ≤ 21(*V* ⋅ *mV*), 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.

## IV. CONCLUSIONS

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 *Q* − *V*_{dc} 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 *Q* − *V*_{dc} 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 10^{5} 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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors declare no conflict of interest.

### Author Contributions

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.

## DATA AVAILABILITY

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

### APPENDIX A: MODEL OF CIRCULAR GRAPHENE RESONATOR

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

where $\beta 10$ 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

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

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

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

where $a=\beta 1(0)R$.

### APPENDIX B: FITTING PARAMETERS OF FIG. 2(b)

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*. $ab=\sigma min\epsilon 0\mu /d0$, where *σ*_{min} is the minimum conductivity of graphene, and on the order of tens of *μS*; *ɛ*_{0} = 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 10^{4} ∼ 10^{5} cm^{2}/V s at low temperature; *d*_{0} 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°. $cd=112Vac2Vdc2D22d02\pi 2$, where *V*_{ac} is the AC gate voltage, which ranges from tens to hundreds of mW. *V*_{dc} 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^{−9}*m*. Therefore, it can be obtained that *c*/*d* is reasonable on the order of 10^{2} ∼ 10^{6}. 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.