Modulation of Mechanical Resonance by Chemical Potential Oscillation in Graphene

LETTERS PUBLISHED ONLINE: 7 DECEMBER 2015 | DOI: 10.1038/NPHYS3576

Modulation of mechanical resonance by chemical potential oscillation in graphene

Changyao Chen1†, Vikram V. Deshpande2, Mikito Koshino3, Sunwoo Lee4, Alexander Gondarenko1, Allan H. MacDonald5, Philip Kim6 and James Hone1*

The classical picture of the force on a capacitor assumes a investigated. Understanding the corrections to the classical model large density of electronic states, such that the electrochemical is of fundamental interest and important for modelling of nano- potential of charges added to the capacitor is given by mechanical systems in the atomically thin limit3,4. Here we examine the external electrostatic potential and the capacitance is the case of a resonant nano-mechanical device, and show that the determined purely by geometry1. Here we consider capacitively first-order correction to the resonant frequency consists of three driven motion of a nano-mechanical resonator with a low terms: one term proportional to µ and the other two proportional density of states, in which these assumptions can break to its derivative dµ/dn, with the latter represented through the in- 2–5 −1 =( / 2)( µ/ down . We find three leading-order corrections to the verse quantum capacitance CQ 1 Ae d dn)(A is the sample classical picture: the first of which is a modulation in the size and e is the electron charge). The phenomenon is explored static force due to variation in the internal chemical potential; experimentally by examining the behaviour of a graphene resonator, the second and third are changes in the static force and in which µ is tuned by a perpendicular magnetic field B at a dynamic spring constant due to the rate of change of chemical fixed electrochemical potential. We observe strong frequency shifts, potential, expressed as the quantum (density of states) which are periodic in 1/B, and can be quantitatively described by the capacitance6,7. As a demonstration, we study capacitively theoretical model. We further find that, in the limit of an ultra-clean driven graphene mechanical resonators, where the chemical device with low tension, the first term can provide the dominant potential is modulated independently of the gate voltage using tuning effect, demonstrating that a nano-mechanical device can be an applied magnetic field to manipulate the energy of electrons used to track its own chemical potential. residing in discrete Landau levels8–10. In these devices, we We model a generic mechanical resonator as a mass on a observe large periodic frequency shifts consistent with the nonlinear spring, capacitively actuated by a nearby gate electrode, as three corrections to the classical picture. In devices with shown in Fig.1. A direct current (d.c.) voltage Vg applied to the gate extremely low strain and disorder, the first correction term can both increase the resonant frequency by exerting a static force dominates and the resonant frequency closely follows the to change the equilibrium deflection ze, and decrease the frequency chemical potential. The theoretical model fits the data with only owing to the nonlinear electrostatic potential, an effect known as one adjustable parameter representing disorder-broadening of electrostatic spring softening15. The two effects provide an effective the Landau levels. The underlying electromechanical coupling spring constant given by: mechanism is not limited by the particular choice of material, 2 geometry, or mechanism for variation in the chemical potential, 1 d Ctotal k =k+ηαz 2 − V 2 (1) and can thus be extended to other low-dimensional systems. eff e 2 dz 2 g The calculation of the force between plates of a parallel-plate capacitor by energy methods is a classic problem that illustrates the where k is the linear spring constant, α is the elastic nonlinear importance of the correct definition of the system’s free energy: con- coefficient, and η is the geometric coefficient, which depends on sidering only the energy stored in the capacitor incorrectly predicts the exact form of static deflection (see Supplementary Section 2). In a repulsive force of (1/2)(dC/dz)φ2, where C is the capacitance, rigid microelectromechanical devices (MEMS), the last (capacitive dC/dz is its spatial derivative, and φ is the electrostatic potential softening) term provides the dominant tuning effect, whereas in difference. The correct result (an attractive force −(1/2)(dC/dz)φ2) atomically thin materials under low tension, the second (mechanical is obtained only when the work done by the battery to maintain a stiffening) term can dominate. constant voltage is taken into account1. This result assumes a simple The role of the finite density of states is modelled by considering system of metallic capacitors with a large density of states (DOS), the variation in µ with carrier density n, such that the electrostatic φ µ/ such that the chemical potential µ is constant and the electrostatic potential is given by =Vg − e, and by including CQ in series with potential is identical to the voltage applied by the battery. This Cg. In this case, we first find that the static capacitive force is given assumption breaks down in nanoscale systems, in which the DOS by (Supplementary Section 3): can be much smaller and µ is no longer fixed, an effect widely µ 2 studied in electronic devices6,7,11–14. However, the role of a finite 1 Cg dCg F =− Vg − 1−2 (2) DOS in the force between capacitor plates has not been rigorously 2 e CQ dz

1Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA. 2Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA. 3Department of Physics, Tohoku University, Sendai 980-8578, Japan. 4Department of Electrical Engineering, Columbia University, New York, New York 10027, USA. 5Department of Physics, University of Texas, Austin, Texas 78712, USA. 6Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. †Present address: Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA. *e-mail: [email protected]

three-terminal configuration (Fig.2a) with a local gate (LG) placed few hundred nanometres away from the suspended graphene. The ε graphene is assumed to have built-in strain 0 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 kz + αz3 ε at different levels of 0 (Fig.2b, also Supplementary Section 3). ε μ Figure2c shows oscillations), and in the magnetization M =−dµ/dB (de Haas–van C and the electrostatic potential φ, but also by the quantum capacitance g Alphen oscillations, typically detected by torque magnetometry2), CQ. For fixed electrochemical potential Vg, φ is directly modified by the µ but can also be studied explicitly using a sensitive single- chemical potential , therefore the total spring constant ktotal is modulated 18,19 µ α electron transistor electrometer . In a single-particle picture, by both and CQ, and magnified by the nonlinear eect (large ). µ the dependence of and CQ on B is determined by the filling fraction ν = 2πnh¯/eB (h¯ is the reduced Planck’s constant) and µ 20 When Vg is kept constant and and CQ are modulated, the change disorder. We construct a simple model consisting of up to in static force is approximated as: 20 Gaussian-broadened√ LLs with disorder-induced width Γ (in 6 −1 units of vF 2ehB¯ , where vF = 10 m s is the Fermi velocity). The 1µ ( ) = / µ ∝ PN [−( − )2/Γ 2] 1 1 1 dCg DOS is given by D E dn d exp E EN , where F ≈− +CgVg Vg (3) e C dz EN is the centre of the Nth LL. Within each LL,√EN of single-layer Q ( ) ( ) µ graphene evolves with B as EN B = vFsgn N 2ehB¯ |N|. Here The change in static force shifts the equilibrium deflection ze— is assigned as the highest filled energy level, and CQ is defined and thus the resonant frequency—and alters the curvature of the as Ae2D(E), where A is the sample area. Figure2d–f shows the potential energy at the equilibrium position, as determined from the simulated 1µ as a function of B and accompanying frequency shifts spatial derivative of 1F. The leading term in the latter is given by: calculated from each of the terms in equation (5) for three different combinations of high/low tension and disorder. In the case of low 1 2 1 1 d F dCg 1 1 tension but large disorder (Fig.2d), f1,2,3 are all reflected in the total k= = Vg (4) dz dz CQ frequency shift, but LL broadening largely obscures the variation in µ. In the case of high tension and small disorder (Fig.2e), we found 1 > 1 Combining the effects from equations (2) and (3), we arrive at the that f3 f1,2 as expected, leading to sharp spikes in frequency total frequency shift as: at LL transitions. Finally, in the case of both low tension and low disorder (Fig.2f), the frequency shift closely (but not completely) 1µ 1 1 µ 1 =−< −< 1 follows the contribution of f1 and the sawtooth variation of , with ftotal V VCgVg (5) 1 e CQ spikes at the transitions between LLs due to f2,3. This simulation motivates the use of graphene samples with very f dC 2 1 low built-in strain and disorder, to directly observe variation in µ. + 0 g 1 Vg To prevent contamination, we prepare samples by direct exfoliation 2k0 dz CQ of graphite over pre-patterned electrodes21, and further clean the / 22,23 where

ab c 200 15 S D LG Detection ) 10 150 −1 5 (MHz V

100 V 0 Vg (V) 1 Frequency (MHz) ε = 0.1% 5 50 0 −5 ε = 0.01% 10 Vg 0 −10 −5 0 5 10 0.001 0.01 0.1 1 ε Vg (V) Initial strain, 0 (%)

(meV) − (meV) − (meV) −

μ μ de−1 −60 μ 10 −1 −50 f100 −1 −50 20 V = 11.1 MHz V V = 1.1 MHz V V = 11.1 MHz V Γ Γ Γ = 0.25 Δ = 0.125 Δ −60 = 0.125 Δ −60 f1 0 f1 0 f1 −61 −70 −70 0 Δ Δ f3 Δf 100 f2 3,000 2,000 Δ 3 min min

min Δ f

f f f 2 0 f 1,500 3 Δ Δ Δ Δf 0 / /

/ 2 f f −100 f 0 Δ Δ Δ −2,000 Δ Δ ftotal 10 ftotal Δf 1,000 total 100 −10 0 0

0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 B (T) B (T) B (T)

Figure 2 | Mechanical resonance modulation. 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 dierent initial strain ε0. c, Calculated

There is no obvious frequency shift with B, except for regimes both the third term in equation (5) and the classical electrostatic between LLs, where sharp spikes are evident but the signal softening term in equation (1) should decrease with the decreasing becomes weak. Such observations agree well with the predictions dynamic capacitance; in general, the classical term will dominate for small

a a × 10−5 −1 123.7 |S21| Device D1, V = 2 MHz V 116.0 05 Vg = −7.5 V N = 2 Vg = −6 V

123.5 115.6 Frequency (MHz) 123.3 106.0 Vg = −6.5 V N = 2 246 8 B (T) 105.5

−20 Frequency (MHz) b −1 75.8 Device D2, V = 10 MHz V 65.5 Vg = −4.2 V Vg = −3.4 V (meV) −

N = 1 −40 μ 65.0 N = 1 75.4 N = 5

−60 Frequency (MHz) ∠S (°) N = 2 75.0 21 −100 100 N = 3 0123 012N = 4 B (T) B (T) b 10 −7.5 V c −6.5 V 75.8 5 −4.2 V −3.4 V

75.6 (meV)

0 0

75.4 − μμ −5 75.2 Frequency (MHz) −10 2 6 10 14 18 22 012 Filling factor, ν B (T) cd80 20 −7.5 V Figure 3 | Chemical-potential-variation-induced frequency shifts. −6.5 V a, Measured magnitude of S21 transmission as a function of applied −4.2 V −3.4 V 60 magnetic field for device D1. Owing to the small frequency tunability, there (meV) (meV) 10 is no obvious mechanical resonance shift within a single LL. Test conditions: μ 0 μ Δ 40 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 0 20 6101418 2468 show the LL energies for N=1–5. Test conditions: T =4.3 K, Vg =−4.2 V, Filling factor, ν V (V) drive power is −68 dB m. c, Complete fitted result (red curve) to the data g shown in b. The dashed line indicates where the model is not expected to be accurate. Figure 4 | Chemical potential evolutions and overall fits to experimental data. a, Mechanical resonant frequency as functions of the applied Our nano-mechanical technique thus provides access to a key magnetic fields at dierent Vg, and corresponding fits (red curves). thermodynamic quantity (chemical potential) in a self-contained The dashed lines indicate where the model is not expected to be accurate manner. Specifically, our system allows measurements of both (see text). b, Extracted chemical potential changes for dierent Vg, as a the energy gaps and the gradual energy transition between gaps, function of the filling factors. The dashed blue line shows the linear which would otherwise require specialized instrumentation18,19. extrapolation used to determine the energy gaps at integer filling factors. In the future, measurements of other correlation effects such as c, Energy gaps at dierent integer filling factors for various Vg. d, Chemical 28,29 19,30–34 µ negative compressibility , fractional quantum Hall effects potential at zero magnetic field, 0 at dierent Vg. Red solid curve is a 35 p and Wigner crystallization , all of which have thermodynamic Vg fit. signatures, should be possible using this technique. Finally, we emphasize that the electromechanical mechanism identified in the References present work is not specific to graphene. Indeed, any atomically 1. Feynman, R. P., Leighton, R. B. & Sands, M. The Feynman Lectures on Physics thin mechanical resonator with a quasi-2D electronic structure Vol. 2 (Basic Books, 2013). will demonstrate this sensitive response to external fields via an 2. Eisenstein, J. P. et al. Density of states and de Haas–van Alphen effect in electrostatically controlled mechanism. two-dimensional electron systems. Phys. Rev. Lett. 55, 875–878 (1985). 3. Steele, G. A. et al. Strong coupling between single-electron tunneling and nanomechanical motion. Science 325, 1103–1107 (2009). Methods 4. Lassagne, B., Tarakanov, Y., Kinaret, J., Garcia-Sanchez, D. & Bachtold, A. Methods and any associated references are available in the online Coupling mechanics to charge transport in carbon nanotube mechanical version of the paper. resonators. Science 325, 1107–1110 (2009). 5. Singh, V. et al. Coupling between quantum Hall state and electromechanics in Received 8 July 2015; accepted 27 October 2015; suspended graphene resonator. Appl. Phys. Lett. 100, 233103 (2012). 6. Ilani, S., Donev, L. A., Kindermann, M. & McEuen, P. L. Measurement of the published online 7 December 2015; corrected online quantum capacitance of interacting electrons in carbon nanotubes. Nature 25 February 2016 Phys. 2, 687–691 (2006).

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Methods Device characterization. Suspended graphene samples are electrically tested at Device fabrication. High-resistivity silicon wafers (>20,000 cm) are used in the room temperature under vacuum before cool-down. Only samples that possess experiments described in the main text, to minimize the parasitic capacitance in reasonable transconductance (1R>100 within 10 V change of gate voltage) the radiofrequency (RF) range. 290 nm of thermal oxide is then grown to have the are chosen for further investigation. best optical contrast for identifying single-layer graphene in later studies. For low-temperature measurements in a magnetic field, samples are Source and drain electrodes (1 nm Cr, 15 nm Au) are first patterned by deep wire-bonded and loaded into an in-house-built insert for the Quantum ultraviolet photo-lithography (ASML 300C DUV stepper, Cornell Nanofabrication Design physical properties measurement system (PPMS, 1.8 K base

Facility). Next, a dry etch (CF4) creates 200-nm-deep trenches between source and temperature) with eight semi-rigid coaxial cables to transmit both d.c. and drain electrodes, and finally local gate electrodes (1 nm Cr, 20 nm Au) are RF signals. Most suspended graphene samples show improved quality deposited in the trenches with careful alignment. The details of the fabrication are (narrower Dirac peak) on cooling. Sample quality is further described elsewhere24. improved by gentle current annealing23. After such treatments, the smallest Graphene samples are prepared by mechanical exfoliation of kish graphite. The full-width at half-maximum (FWHM) of the Dirac peak is about 0.2 V, colour contrast of the substrate-supported region aids in identification of thin corresponding to a disorder density of ∼8×109 cm−2. To avoid the graphene flakes crossing the electrodes. The precise number of layers is subsequently collapsing as a result of electrostatic forces, we apply a gate voltage only confirmed by Raman spectroscopy. within 10 V.

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