Modulation of mechanical resonance by chemical potential oscillation in graphene

Journal name:
Nature Physics
Volume:
12,
Pages:
240–244
Year published:
DOI:
doi:10.1038/nphys3576
Received
Accepted
Published online
Corrected online

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 geometry1. Here we consider capacitively driven motion of a nano-mechanical resonator with a low density of states, in which these assumptions can break down2, 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) capacitance6, 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 levels8, 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.

At a glance

Figures

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

    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).

  2. Mechanical resonance modulation.
    Figure 2: Mechanical resonance modulation.

    a, Schematic of the graphene mechanical resonator with source (S), drain (D) and local gate (LG) contacts. Scale bar, 1μm. b, Simulated resonant frequency for a 2 by 2μm graphene resonator with a 120nm gap, at different initial strain ε0. c, Calculated as a function of initial strain ε0 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 = 10V,ε0 = 0.01%,Q = 2,000, signal-to-noise ratio = 10dB,f0 = 161.4MHz, and fmin is 360Hz assuming a 100Hz measurement bandwidth.

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

    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 = 4.3K,Vg = −6V, drive power is −62dBm. 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 = 4.3K,Vg = −4.2V, drive power is −68dBm. 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.

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

    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.

Change history

Corrected online 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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Author information

  1. Present address: Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA.

    • Changyao Chen

Affiliations

  1. Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA

    • Changyao Chen,
    • Alexander Gondarenko &
    • James Hone
  2. Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA

    • Vikram V. Deshpande
  3. Department of Physics, Tohoku University, Sendai 980-8578, Japan

    • Mikito Koshino
  4. Department of Electrical Engineering, Columbia University, New York, New York 10027, USA

    • Sunwoo Lee
  5. Department of Physics, University of Texas, Austin, Texas 78712, USA

    • Allan H. MacDonald
  6. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    • Philip Kim

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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The authors declare no competing financial interests.

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