## Abstract

The classical picture of the force on a capacitor assumes a large density of electronic states, such that the electrochemical potential of charges added to the capacitor is given by the external electrostatic potential and the capacitance is determined purely by geometry^{1}. Here we consider capacitively driven motion of a nano-mechanical resonator with a low density of states, in which these assumptions can break down^{2,3,4,5}. We find three leading-order corrections to the classical picture: the first of which is a modulation in the static force due to variation in the internal chemical potential; the second and third are changes in the static force and dynamic spring constant due to the rate of change of chemical potential, expressed as the quantum (density of states) capacitance^{6,7}. As a demonstration, we study capacitively driven graphene mechanical resonators, where the chemical potential is modulated independently of the gate voltage using an applied magnetic field to manipulate the energy of electrons residing in discrete Landau levels^{8,9,10}. In these devices, we observe large periodic frequency shifts consistent with the three corrections to the classical picture. In devices with extremely low strain and disorder, the first correction term dominates and the resonant frequency closely follows the chemical potential. The theoretical model fits the data with only one adjustable parameter representing disorder-broadening of the Landau levels. The underlying electromechanical coupling mechanism is not limited by the particular choice of material, geometry, or mechanism for variation in the chemical potential, and can thus be extended to other low-dimensional systems.

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## Main

The calculation of the force between plates of a parallel-plate capacitor by energy methods is a classic problem that illustrates the importance of the correct definition of the system’s free energy: considering only the energy stored in the capacitor incorrectly predicts a repulsive force of (1/2)(d*C*/d*z*)*φ*^{2}, where *C* is the capacitance, d*C*/d*z* is its spatial derivative, and *φ* is the electrostatic potential difference. The correct result (an attractive force −(1/2)(d*C*/d*z*)*φ*^{2}) is obtained only when the work done by the battery to maintain a constant voltage is taken into account^{1}. This result assumes a simple system of metallic capacitors with a large density of states (DOS), such that the chemical potential *μ* is constant and the electrostatic potential is identical to the voltage applied by the battery. This assumption breaks down in nanoscale systems, in which the DOS can be much smaller and *μ* is no longer fixed, an effect widely studied in electronic devices^{6,7,11,12,13,14}. However, the role of a finite DOS in the force between capacitor plates has not been rigorously investigated. Understanding the corrections to the classical model is of fundamental interest and important for modelling of nano-mechanical systems in the atomically thin limit^{3,4}. Here we examine the case of a resonant nano-mechanical device, and show that the first-order correction to the resonant frequency consists of three terms: one term proportional to *μ* and the other two proportional to its derivative d*μ*/d*n*, with the latter represented through the inverse quantum capacitance *C*_{Q}^{−1} = (1/*Ae*^{2})(d*μ*/d*n*) (*A* is the sample size and *e* is the electron charge). The phenomenon is explored experimentally by examining the behaviour of a graphene resonator, in which *μ* is tuned by a perpendicular magnetic field *B* at a fixed electrochemical potential. We observe strong frequency shifts, which are periodic in 1/*B*, and can be quantitatively described by the theoretical model. We further find that, in the limit of an ultra-clean device with low tension, the first term can provide the dominant tuning effect, demonstrating that a nano-mechanical device can be used to track its own chemical potential.

We model a generic mechanical resonator as a mass on a nonlinear spring, capacitively actuated by a nearby gate electrode, as shown in Fig. 1. A direct current (d.c.) voltage *V*_{g} applied to the gate can both increase the resonant frequency by exerting a static force to change the equilibrium deflection *z*_{e}, and decrease the frequency owing to the nonlinear electrostatic potential, an effect known as electrostatic spring softening^{15}. The two effects provide an effective spring constant given by:

where *k* is the linear spring constant, *α* is the elastic nonlinear coefficient, and *η* is the geometric coefficient, which depends on the exact form of static deflection (see Supplementary Section 2). In rigid microelectromechanical devices (MEMS), the last (capacitive softening) term provides the dominant tuning effect, whereas in atomically thin materials under low tension, the second (mechanical stiffening) term can dominate.

The role of the finite density of states is modelled by considering the variation in *μ* with carrier density *n*, such that the electrostatic potential is given by *φ* = *V*_{g} − *μ*/*e*, and by including *C*_{Q} in series with *C*_{g}. In this case, we first find that the static capacitive force is given by (Supplementary Section 3):

When *V*_{g} is kept constant and *μ* and *C*_{Q} are modulated, the change in static force is approximated as:

The change in static force shifts the equilibrium deflection *z*_{e}—and thus the resonant frequency—and alters the curvature of the potential energy at the equilibrium position, as determined from the spatial derivative of Δ*F*. The leading term in the latter is given by:

Combining the effects from equations (2) and (3), we arrive at the total frequency shift as:

where indicates how easily the resonant frequency *f*_{0} can be tuned by electrostatic potential, *k*_{0} is the unmodulated spring constant, Δ*μ* is the change in the chemical potential, and Δ(1/*C*_{Q}) represents the change in the inverse quantum capacitance. The three terms in equation (5) are referred to below as Δ*f*_{1}, Δ*f*_{2} and Δ*f*_{3}, respectively. Importantly Δ*f*_{1} is a direct measure of variation in the chemical potential, whereas Δ*f*_{2} and Δ*f*_{3} depend on the change in quantum capacitance. We also note that these terms are closely related to experiments with nanotube quantum dot mechanical resonators^{3,4}, in which Δ*f*_{2,3} are analogous to the diverging charge susceptibility of the quantum dot that produces dips in the nanotube mechanical frequency at charge transitions, and Δ*f*_{1} plays a similar role that changes the static tension and the mechanical frequency from discretely charging the quantum dot.

We examine this phenomenon experimentally using graphene mechanical resonators^{16,17}, where a transverse magnetic field *B* is used to tune *μ* and *C*_{Q} independently of *V*_{g}. The samples are in a three-terminal configuration (Fig. 2a) with a local gate (LG) placed few hundred nanometres away from the suspended graphene. The graphene is assumed to have built-in strain *ɛ*_{0} from the fabrication process, with additional strain induced by the d.c. gate voltage *V*_{g}. We use a continuum mechanical model for the calculation of *f*(*V*_{g}) at different levels of *ɛ*_{0} (Fig. 2b, also Supplementary Section 3). Figure 2c shows as a function of *ɛ*_{0}, at three different values of *V*_{g}, for a 2 by 2 μm single-layer graphene resonator. Thus, in devices with low *ɛ*_{0}, can be large and Δ*f*_{1,2} are maximized, whereas in devices with large *ɛ*_{0}, is substantially smaller and Δ*f*_{1,2} should be minimized relative to Δ*f*_{3}. We note that previous studies of graphene resonators in high magnetic fields^{5} utilized samples with high tension and, therefore, observed only the effects of Δ*f*_{3}; in the experimental work below, we focus on samples with ultralow tension to directly observe the effects of chemical potential variation.

In a transverse magnetic field, electronic states of graphene form discrete Landau levels (LLs), and *μ* oscillates in a sawtooth pattern with increasing *B*. The oscillation in *μ* is reflected in the longitudinal electrical resistivity^{8,9,10} (Shubnikov–de Haas oscillations), and in the magnetization *M* = −d*μ*/d*B* (de Haas–van Alphen oscillations, typically detected by torque magnetometry^{2}), but can also be studied explicitly using a sensitive single-electron transistor electrometer^{18,19}. In a single-particle picture, the dependence of *μ* and *C*_{Q} on *B* is determined by the filling fraction *ν* = 2π*nℏ*/*eB* (*ℏ* is the reduced Planck’s constant) and disorder. We construct a simple model^{20} consisting of up to 20 Gaussian-broadened LLs with disorder-induced width *Γ* (in units of , where *v*_{F} = 10^{6} m s^{−1} is the Fermi velocity). The DOS is given by *D*(*E*) = d*n*/d*μ* ∝ ∑ ^{N} exp[−(*E* − *E*_{N})^{2}/*Γ*^{2}], where *E*_{N} is the centre of the *N*th LL. Within each LL, *E*_{N} of single-layer graphene evolves with *B* as . Here *μ* is assigned as the highest filled energy level, and *C*_{Q} is defined as *Ae*^{2}*D*(*E*), where *A* is the sample area. Figure 2d–f shows the simulated Δ*μ* as a function of *B* and accompanying frequency shifts calculated from each of the terms in equation (5) for three different combinations of high/low tension and disorder. In the case of low tension but large disorder (Fig. 2d), Δ*f*_{1,2,3} are all reflected in the total frequency shift, but LL broadening largely obscures the variation in *μ*. In the case of high tension and small disorder (Fig. 2e), we found that Δ*f*_{3} > Δ*f*_{1,2} as expected, leading to sharp spikes in frequency at LL transitions. Finally, in the case of both low tension and low disorder (Fig. 2f), the frequency shift closely (but not completely) follows the contribution of Δ*f*_{1} and the sawtooth variation of *μ*, with spikes at the transitions between LLs due to Δ*f*_{2,3}.

This simulation motivates the use of graphene samples with very low built-in strain and disorder, to directly observe variation in *μ*. To prevent contamination, we prepare samples by direct exfoliation of graphite over pre-patterned electrodes^{21}, and further clean the graphene by Joule heating in vacuum at low temperature^{22,23}. The electronic quality is examined through separate measurements (Supplementary Section 1), and can yield a charged impurity density as low as ∼8 × 10^{9} cm^{−2}. For mechanical resonance measurements, the graphene is actuated electrostatically by adding a small radiofrequency drive voltage to the gate, and the gate–drain current at the same frequency is read out by a vector network analyser; on resonance, the changing capacitance due to mechanical motion causes a measurable peak in the current^{21,24}. The absence of d.c. bias avoids Joule heating and possible Lorentz forces. We scan *V*_{g} up to ±10 V at zero magnetic field to determine and use the continuum mechanical model to extract *ɛ*_{0}. The use of substrate-fixed electrodes eliminates tensioning owing to thermal contraction of the metal^{17,25}, and we are able to achieve as large as 10 MHz V^{−1}, with *ɛ*_{0} ∼ 10^{−4}.

Figure 3a shows the response of the mechanical resonance to the applied magnetic field *B*, for device D1 with of 2 MHz V^{−1}. There is no obvious frequency shift with *B*, except for regimes between LLs, where sharp spikes are evident but the signal becomes weak. Such observations agree well with the predictions for small and experimental results reported by Singh and colleagues. Figure 3b shows the same measurements on device D2 (length 2.4 μm, width 3.2 μm), with of 10 MHz V^{−1}. The data reveal a repeated pattern of oscillations that are periodic in 1/*B* (Supplementary Section 4), allowing us to directly extract the carrier density *n* of 2.4 × 10^{11} cm^{−2}. We use the measured values of *n* at five values of *V*_{g} to obtain *C*_{g}. Finite element simulation (COMSOL Multiphysics) is then used to obtain the effective displacement *z*_{e}, and d*C*_{g}/d*z*_{e} is approximated as *C*_{g}/*z*_{e}. Therefore, the only free parameter left to fit equation (5) is the disorder level *Γ*, which will determine Δ*μ* and Δ(1/*C*_{Q}). We found that a *Γ* of 0.1 gives a satisfactory result. The corresponding *μ*(*B*), that gives rise to Δ*f*_{1}, is overlaid as a yellow curve in Fig. 3b. For comparison, we also plot *E*_{N}(*B*) for the first five LLs as dotted green lines. We found that *μ* traces *E*_{N} closely in each LL, confirming that the frequency can closely track *μ* in high-quality samples. The detailed fit between the model and the data is shown in Fig. 3c. Here, the circles represent the resonant frequency extracted at each value of *B* from the data in Fig. 3b, and the line represents the prediction of equation (5) with *Γ* = 0.1. The fit is excellent and validates the model presented above.

We briefly consider the region between LLs, where the frequency shift should be primarily sensitive to *C*_{Q}. At high magnetic fields, when the sample enters the well-developed quantum Hall regime, its bulk becomes insulating and the edges host dissipationless one-dimensional edge states. This will not change the static charge on the sample and the static force due to *V*_{g}, and therefore first two terms of equation (5) should be unaffected. However, when the bulk becomes sufficiently insulating such that its RC charging time is greater than the mechanical resonance period, the dynamic charging should be determined by the geometric capacitance between the gate and the edge channels, which can be an order of magnitude smaller than *C*_{g} (see Supplementary Section 5). This has two consequences. First, both the third term in equation (5) and the classical electrostatic softening term in equation (1) should decrease with the decreasing dynamic capacitance; in general, the classical term will dominate and the frequency spike will be larger than predicted by equation (5). Second, both the radiofrequency drive and capacitive displacement current will also decrease, leading to substantial loss of signal. As a consequence, we can use the loss of signal to directly determine the onset of the bulk insulating behaviour. In Fig. 3b, c (also see Supplementary Fig. 11), this is seen to occur for the final spike, at the transition between *N* = 2 and *N* = 1, indicated by the dashed line in Fig. 3c. For this transition, we expect that the model will not fit the data well; indeed, the magnitude and width of the frequency spike, where observable, seem to exceed the prediction. Nevertheless, the best fit of the limited data in the region between LLs yields *C*_{Q} with values similar to previous measurements of high-quality graphene samples^{26,27}. More detailed modelling and alternative transduction techniques (for example, optical detection) may allow better study of the detailed behaviour between LLs.

Figure 4a shows similar data at different values of *V*_{g}, with fits using *Γ* values within ±10%. Because we are able to ‘read off’ chemical potential variation directly through mechanical resonance shifts, we can then track the chemical potential as a function of the filling fraction *ν*. Figure 4b shows Δ*μ*(*ν*) at four different values of *V*_{g}: there are distinct jumps of *μ* at *ν* = 4(*N* + 1/2), which represent the energy gaps between neighbouring LLs. We linearly extrapolate to extract the energy gaps, as plotted in Fig. 4c. In addition, the same analysis allows extrapolation to determine the chemical potential at zero magnetic field . As shown in Fig. 4d, this follows as expected. Therefore, the mechanical measurements can, under appropriate conditions, provide a direct means to monitor the chemical potential evolution of the underlying system, in a straightforward manner. We expect to extend our measurements into extreme quantum regimes at larger *B*, where additional chemical potential jumps would suggest the formation of many-body incompressible states.

Our nano-mechanical technique thus provides access to a key thermodynamic quantity (chemical potential) in a self-contained manner. Specifically, our system allows measurements of both the energy gaps and the gradual energy transition between gaps, which would otherwise require specialized instrumentation^{18,19}. In the future, measurements of other correlation effects such as negative compressibility^{28,29}, fractional quantum Hall effects^{19,30,31,32,33,34} and Wigner crystallization^{35}, all of which have thermodynamic signatures, should be possible using this technique. Finally, we emphasize that the electromechanical mechanism identified in the present work is not specific to graphene. Indeed, any atomically thin mechanical resonator with a quasi-2D electronic structure will demonstrate this sensitive response to external fields via an electrostatically controlled mechanism.

## Methods

### Device fabrication.

High-resistivity silicon wafers (>20,000 Ω cm) are used in the experiments described in the main text, to minimize the parasitic capacitance in the radiofrequency (RF) range. 290 nm of thermal oxide is then grown to have the best optical contrast for identifying single-layer graphene in later studies.

Source and drain electrodes (1 nm Cr, 15 nm Au) are first patterned by deep ultraviolet photo-lithography (ASML 300C DUV stepper, Cornell Nanofabrication Facility). Next, a dry etch (CF_{4}) creates 200-nm-deep trenches between source and drain electrodes, and finally local gate electrodes (1 nm Cr, 20 nm Au) are deposited in the trenches with careful alignment. The details of the fabrication are described elsewhere^{24}.

Graphene samples are prepared by mechanical exfoliation of kish graphite. The colour contrast of the substrate-supported region aids in identification of thin flakes crossing the electrodes. The precise number of layers is subsequently confirmed by Raman spectroscopy.

### Device characterization.

Suspended graphene samples are electrically tested at room temperature under vacuum before cool-down. Only samples that possess reasonable transconductance (Δ*R* > 100 Ω within ±10 V change of gate voltage) are chosen for further investigation.

For low-temperature measurements in a magnetic field, samples are wire-bonded and loaded into an in-house-built insert for the Quantum Design physical properties measurement system (PPMS, 1.8 K base temperature) with eight semi-rigid coaxial cables to transmit both d.c. and RF signals. Most suspended graphene samples show improved quality (narrower Dirac peak) on cooling. Sample quality is further improved by gentle current annealing^{23}. After such treatments, the smallest full-width at half-maximum (FWHM) of the Dirac peak is about 0.2 V, corresponding to a disorder density of ∼8 × 10^{9} cm^{−2}. To avoid the graphene collapsing as a result of electrostatic forces, we apply a gate voltage only within ± 10 V.

## Change history

### 25 February 2016

In the version of this Letter originally published, dashed lines indicating where the model is not expected to be accurate were omitted from the lower two panels in Fig. 4a. This has been corrected in all versions of the Letter.

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## Acknowledgements

The authors thank N. Cooper, I. Aleiner, B. Skinner and G. Steele for helpful discussions; D. Heinz and A. Young for help in building the measurement set-up; N. Clay for fabrication support; K.-C. Fong, T. Heinz, A. Young and A.van der Zande for helpful comments. P.K. and J.H. acknowledge Air Force Office of Scientific Research Grant No. MURI FA955009-1-0705. A.H.M. was supported by the DOE Division of Materials Sciences and Engineering under Grant DE-FG03-02ER45958, and by the Welch Foundation under Grant TBF1473. P.K. acknowledges support from DOE (DE-FG02-05ER46215) for performing experiments and data analysis.

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### Contributions

C.C. and V.V.D. fabricated and characterized the samples, developed the measurement technique and performed the experiments, data analysis and theoretical modelling. M.K. performed disorder-broadened calculations. S.L. and A.G. helped in sample fabrication. A.H.M. provided theoretical support. P.K. and J.H. oversaw the project. C.C., V.V.D., P.K. and J.H. co-wrote the paper. All authors discussed and commented on the manuscript.

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Chen, C., Deshpande, V., Koshino, M. *et al.* Modulation of mechanical resonance by chemical potential oscillation in graphene.
*Nature Phys* **12**, 240–244 (2016). https://doi.org/10.1038/nphys3576

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DOI: https://doi.org/10.1038/nphys3576