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)(dC/dz)φ2, where C is the capacitance, dC/dz is its spatial derivative, and φ is the electrostatic potential difference. The correct result (an attractive force −(1/2)(dC/dz)φ2) is obtained only when the work done by the battery to maintain a constant voltage is taken into account1. 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 devices6,7,11,12,13,14. 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 limit3,4. 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 dμ/dn, with the latter represented through the inverse quantum capacitance CQ−1 = (1/Ae2)(dμ/dn) (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 Vg applied to the gate can both increase the resonant frequency by exerting a static force to change the equilibrium deflection ze, and decrease the frequency owing to the nonlinear electrostatic potential, an effect known as electrostatic spring softening15. 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.

Figure 1: Mass on a nonlinear spring balanced with electrostatic force.
figure 1

The force exerted by the spring varies nonlinearly with the displacement z, and the electrostatic force is determined not only by the gate capacitance Cg and the electrostatic potential φ, but also by the quantum capacitance CQ. For fixed electrochemical potential Vg,φ is directly modified by the chemical potential μ, therefore the total spring constant ktotal is modulated by both μ andCQ, and magnified by the nonlinear effect (large α).

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 = Vgμ/e, and by including CQ in series with Cg. In this case, we first find that the static capacitive force is given by (Supplementary Section 3):

When Vg is kept constant and μ and CQ are modulated, the change in static force is approximated as:

The change in static force shifts the equilibrium deflection ze—and thus the resonant frequency—and alters the curvature of the potential energy at the equilibrium position, as determined from the spatial derivative of ΔF. 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 f0 can be tuned by electrostatic potential, k0 is the unmodulated spring constant, Δμ is the change in the chemical potential, and Δ(1/CQ) represents the change in the inverse quantum capacitance. The three terms in equation (5) are referred to below as Δf1, Δf2 and Δf3, respectively. Importantly Δf1 is a direct measure of variation in the chemical potential, whereas Δf2 and Δf3 depend on the change in quantum capacitance. We also note that these terms are closely related to experiments with nanotube quantum dot mechanical resonators3,4, in which Δf2,3 are analogous to the diverging charge susceptibility of the quantum dot that produces dips in the nanotube mechanical frequency at charge transitions, and Δf1 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 resonators16,17, where a transverse magnetic field B is used to tune μ and CQ independently of Vg. 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 ɛ0 from the fabrication process, with additional strain induced by the d.c. gate voltage Vg. We use a continuum mechanical model for the calculation of f(Vg) at different levels of ɛ0 (Fig. 2b, also Supplementary Section 3). Figure 2c shows as a function of ɛ0, at three different values of Vg, for a 2 by 2 μm single-layer graphene resonator. Thus, in devices with low ɛ0, can be large and Δf1,2 are maximized, whereas in devices with large ɛ0, is substantially smaller and Δf1,2 should be minimized relative to Δf3. We note that previous studies of graphene resonators in high magnetic fields5 utilized samples with high tension and, therefore, observed only the effects of Δf3; in the experimental work below, we focus on samples with ultralow tension to directly observe the effects of chemical potential variation.

Figure 2: Mechanical resonance modulation.
figure 2

a, Schematic of the graphene mechanical resonator with source (S), drain (D) and local gate (LG) contacts. Scale bar, 1 μm. b, Simulated resonant frequency for a 2 by 2 μm graphene resonator with a 120 nm gap, at different initial strain ɛ0. c, Calculated as a function of initial strain ɛ0 for a single-layer graphene resonator, at different gate voltage Vg. The negative indicates an electrostatic softening effect. df, Simulated frequency shifts under high disorder and large , (d); low disorder and small (e); low disorder and large (f). Top panel: μ (to the right) and corresponding frequency oscillation Δf1 (to the left) as a function of magnetic field B. Middle panel: corresponding frequency shifts Δf2 and Δf3 (see main text). Bottom panel: the total frequency shift as function of B, with Δf1 overlaid on top (black), except for d. For all the simulations, Vg = 10 V,ɛ0 = 0.01%,Q = 2,000, signal-to-noise ratio = 10 dB,f0 = 161.4 MHz, and fmin is 360 Hz assuming a 100 Hz measurement bandwidth.

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 resistivity8,9,10 (Shubnikov–de Haas oscillations), and in the magnetization M = −dμ/dB (de Haas–van Alphen oscillations, typically detected by torque magnetometry2), but can also be studied explicitly using a sensitive single-electron transistor electrometer18,19. In a single-particle picture, the dependence of μ and CQ on B is determined by the filling fraction ν = 2πn/eB ( is the reduced Planck’s constant) and disorder. We construct a simple model20 consisting of up to 20 Gaussian-broadened LLs with disorder-induced width Γ (in units of , where vF = 106 m s−1 is the Fermi velocity). The DOS is given by D(E) = dn/dμ N exp[−(EEN)2/Γ2], where EN is the centre of the Nth LL. Within each LL, EN of single-layer graphene evolves with B as . Here μ is assigned as the highest filled energy level, and CQ is defined as Ae2D(E), where A is the sample area. Figure 2d–f shows the simulated Δμ 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), Δf1,2,3 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 Δf3 > Δf1,2 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 Δf1 and the sawtooth variation of μ, with spikes at the transitions between LLs due to Δf2,3.

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 electrodes21, and further clean the graphene by Joule heating in vacuum at low temperature22,23. The electronic quality is examined through separate measurements (Supplementary Section 1), and can yield a charged impurity density as low as 8 × 109 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 current21,24. The absence of d.c. bias avoids Joule heating and possible Lorentz forces. We scan Vg up to ±10 V at zero magnetic field to determine and use the continuum mechanical model to extract ɛ0. The use of substrate-fixed electrodes eliminates tensioning owing to thermal contraction of the metal17,25, and we are able to achieve as large as 10 MHz V−1, with ɛ0 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 × 1011 cm−2. We use the measured values of n at five values of Vg to obtain Cg. Finite element simulation (COMSOL Multiphysics) is then used to obtain the effective displacement ze, and dCg/dze is approximated as Cg/ze. Therefore, the only free parameter left to fit equation (5) is the disorder level Γ, which will determine Δμ and Δ(1/CQ). We found that a Γ of 0.1 gives a satisfactory result. The corresponding μ(B), that gives rise to Δf1, is overlaid as a yellow curve in Fig. 3b. For comparison, we also plot EN(B) for the first five LLs as dotted green lines. We found that μ traces EN 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.

Figure 3: Chemical-potential-variation-induced frequency shifts.
figure 3

a, Measured magnitude of S21 transmission as a function of applied magnetic field for device D1. Owing to the small frequency tunability, there is no obvious mechanical resonance shift within a single LL. Test conditions: T = 4.3 K,Vg = −6 V, drive power is −62 dB m. b, Similar measurement for device D2, which has a greater frequency tunability. The corresponding chemical potential variation is overlaid in yellow. The green dotted lines show the LL energies for N = 1–5. Test conditions: T = 4.3 K,Vg = −4.2 V, drive power is −68 dB m. c, Complete fitted result (red curve) to the data shown in b. The dashed line indicates where the model is not expected to be accurate.

We briefly consider the region between LLs, where the frequency shift should be primarily sensitive to CQ. 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 Vg, 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 Cg (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 CQ with values similar to previous measurements of high-quality graphene samples26,27. 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 Vg, 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 Δμ(ν) at four different values of Vg: 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.

Figure 4: Chemical potential evolutions and overall fits to experimental data.
figure 4

a, Mechanical resonant frequency as functions of the applied magnetic fields at different Vg, and corresponding fits (red curves). The dashed lines indicate where the model is not expected to be accurate (see text). b, Extracted chemical potential changes for different Vg, as a function of the filling factors. The dashed blue line shows the linear extrapolation used to determine the energy gaps at integer filling factors. c, Energy gaps at different integer filling factors for various Vg. d, Chemical potential at zero magnetic field, μ0 at different Vg. Red solid curve is a  fit.

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 instrumentation18,19. In the future, measurements of other correlation effects such as negative compressibility28,29, fractional quantum Hall effects19,30,31,32,33,34 and Wigner crystallization35, 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.


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 (CF4) 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 elsewhere24.

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 (ΔR > 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 annealing23. 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 × 109 cm−2. To avoid the graphene collapsing as a result of electrostatic forces, we apply a gate voltage only within ± 10 V.