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Modulation of Mechanical Resonance by Chemical Potential Oscillation in Graphene

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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 D. = 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 DkCηαz 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 DVg − 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 D− 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] 240 NATURE PHYSICS | VOL 12 | MARCH 2016 | www.nature.com/naturephysics © 2016 Macmillan Publishers Limited. All rights reserved NATURE PHYSICS DOI: 10.1038/NPHYS3576 LETTERS 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 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 " at different levels of 0 (Fig.2b, also Supplementary Section 3). " μ Figure2c shows <V as a function of 0, at three different values of Vg, for a 2 by 2 µm single-layer graphene resonator. Thus, in " 1 devices with low 0, <V can be large and f1,2 are maximized, (1/2)(dC /dz)φ2(1 − (2C /C )) " g g Q C whereas in devices with large 0, <V is substantially smaller and g eVg 1 1 φ f1,2 should be minimized relative to f3. We note that previous e 5 CQ studies of graphene resonators in high magnetic fields utilized samples with high tension and, therefore, observed only the effects 1 of f3; in the experimental work below, we focus on samples with ultralow tension to directly observe the effects of chemical ∼ μ ktotal f(, CQ) potential variation. In a transverse magnetic field, electronic states of graphene form discrete Landau levels (LLs), and µ oscillates in a sawtooth Figure 1 | Mass on a nonlinear spring balanced with electrostatic force. pattern with increasing B. The oscillation in µ is reflected The force exerted by the spring varies nonlinearly with the displacement z, in the longitudinal electrical resistivity8–10 (Shubnikov–de Haas and the electrostatic force is determined not only by the gate capacitance oscillations), and in the magnetization M D−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 D 2πnhN=eB (hN 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-broadenedp LLs with disorder-induced width Γ (in 6 −1 units of vF 2ehBN , where vF D 10 m s is the Fermi velocity). The 1µ . / D = µ / PN [−. − /2/Γ 2U 1 1 1 dCg DOS is given by D E dn d exp E EN , where F ≈− CCgVg Vg (3) e C dz EN is the centre of the Nth LL. Within each LL,pEN of single-layer Q . / . / µ graphene evolves with B as EN B D vFsgn N 2ehBN jNj. 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.
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