Electrostatically Induced Phononic Crystal

D. Hatanaka, A. Bachtold, and H. Yamaguchi

Phys. Rev. Applied 11, 024024 – Published 8 February 2019

Abstract

The possibility of realizing an electrostatically induced phononic crystal is numerically investigated in an acoustic waveguide based on a graphene sheet that is suspended over periodically arrayed electrodes. The application of dc voltage to these electrodes exerts an electrostatic force on the graphene and this results in the periodic formation of stress in the waveguide structure in a noninvasive way, unlike the cases with mass loading and air holes. This noninvasive scheme enables a band gap, namely a phononic crystal, to be created in the waveguide that can be used to dynamically tune the acoustic transparency in the medium. Our approach will allow the dispersion relation to be locally modified, thus modulating the response of traveling acoustic phonon waves. This alternate phonon architecture is promising in terms of realizing advanced control of phonon dynamics such as waveform engineering.

Received 27 July 2018
Revised 14 December 2018

DOI:https://doi.org/10.1103/PhysRevApplied.11.024024

Physics Subject Headings (PhySH)

Acoustic metamaterials Acoustic phonons Acoustic signal processing & noise Band gap

Graphene Micromechanical & nanomechanical oscillators Waveguides

Sound wave techniques

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

D. Hatanaka^1,*, A. Bachtold², and H. Yamaguchi¹

¹NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan
²ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona) 08860, Spain

^*daiki.hatanaka.hz@hco.ntt.co.jp

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Vol. 11, Iss. 2 — February 2019

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Images

Figure 1
(a) A schematic of a graphene-based acoustic WG where the graphene film is suspended over a trench with a gate electrode array at its bottom. Electrodes with the dimensions $w_{g x} \times w_{g y}$ are periodically arrayed with a distance of $a$ between them. Graphene with a thickness $t$ is suspended by a trench with a width $w$ and is spatially separated from the gate electrodes by a distance $d$ . (b) A unit structure having a length $p$ that is a periodic distance between voltage-biased electrodes. The Floquet boundary condition (BC) is used to calculate the dispersion relation. Perfect matched layers (PMLs) are introduced to avoid undesired reflection.
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Figure 2
(a) A schematic showing the cross section of the unit structure of the graphene WG used for FEM calculations. The unit length along the $x$ axis is $p = 2 a = 6.4 μ m$ . Here, the dc gate voltage is applied between one electrode and the graphene. (b) Electric field distribution and static displacement of the graphene sheet along the $x$ axis. The electric field in the $z$ direction $E_{z}$ is represented on the $x$ - $z$ plane at $y = 0$ when applying $V_{g} = 5$ V. The field $E_{z}$ is strongly confined in a vacuum gap with $d = 85$ nm between the electrode and graphene. This exerts electrostatic force on the graphene and bends it toward the electrode (gray line). (c),(d) The gate-graphene separation $d$ and the gate voltage $V_{g}$ dependence of the maximum static displacement $z_{s}^{\max}$ [see (b)], respectively. (e),(f) The gate-graphene separation $d$ and the gate voltage $V_{g}$ dependence of the stress at clamping edges ( $x$ , $y$ ) = (0, $\pm w / 2$ ), respectively. They are electrostatically induced by $V_{g} = 5$ V [(c) and (e)] and create the separation $d$ = 85 nm [(d),(f)].
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Figure 3
The dispersion relation of the graphene-based WG with $d = 130$ nm at various gate voltages $V_{g} s$ . The dispersion is simulated by applying the Floquet periodic condition to the unit structure shown in Fig. 2. An increase in $V_{g}$ creates the band gaps (yellow) and the reduction in the dispersion slopes (pink circles).
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Figure 4
(a)–(c) The simulated profile of the static displacement $z_{s}$ (left panel) and stress (right panel) in the unit structure of the graphene WG with $d = 130$ nm and $p = a, 2 a$ , and $3 a$ when applying $V_{g} = 8$ V. The static displacement $z_{s}$ in the $x$ - $z$ plane at $y =$ 0 as shown in the bottom panel. The inset in the bottom panel is a schematic of the top view and the $x$ - $y$ - $z$ coordinates. The black dotted frame denotes the unit structure with length $p$ . (d) The eigenfrequencies as a function of the gate voltage $V_{g}$ with $p = a, 2 a$ , and $3 a$ as shown in the top, middle, and bottom panels, respectively. The band gaps are highlighted in yellow. (e) The periodic pitch $a$ dependence of the eigenfrequencies in the device at $V_{g} = 6$ V. The band gap is highlighted in yellow. The black dashed lines denote $a = 3.2$ , 6.4, and 9.6 $μ m$ , where the same configurations are realized as in (a)–(c), respectively.
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Figure 5
(a) A schematic showing the top view of the graphene WG. The suspended membrane part is used for calculation where a fixed BC is applied to the edges of the graphene, denoted as the dotted lines. Harmonic vibrations are locally driven from the left side of the WG through the gate electrode with $V_{d}^{ac}$ and $V_{d}^{dc}$ , and the resulting vibration is calculated at the graphene above the center of the rightmost electrode. To investigate the electrostatic effect on the transmission property, the gate voltage $V_{g}$ is uniformly applied to every two gate electrodes, $p = 2 a$ . An internal loss factor $1 / Q \sim$ 0.01 is used in the simulations as typically measured in graphene resonators at room temperature [29, 33, 38]. (b) The frequency response of the graphene WG at the right edge when exciting the left one with $V_{d}^{ac} = 0.4$ V and $V_{d}^{dc} = 4$ V for different $V_{g}$ values. The vibration amplitudes are suppressed in some frequency ranges (highlighted in gray) due to the band gap created by applying $V_{g}$ .
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