Mode shape and dispersion relation of bending waves in thin silicon membranes

Reimar Waitz, Stephan Nößner, Michael Hertkorn, Olivier Schecker, and Elke Scheer

Phys. Rev. B 85, 035324 – Published 30 January 2012; Erratum Phys. Rev. B 86, 039904 (2012)

Abstract

We study the vibrational behavior of silicon membranes with a thickness of a few hundred nanometers and macroscopic lateral size. A piezo is used to couple in transverse vibrations, which we monitor with a phase-shift interferometer using stroboscopic light. The observed wave pattern of the membrane deflection is a superposition of the mode corresponding to the excitation frequency and several higher harmonics. Using a Fourier transformation in time, we separate these contributions and image up to the eighth harmonic of the excitation frequency. With this method we determine the dispersion relation of membrane oscillations in a frequency range up to 12 MHz. We develop a simple analytical model combining stress of a membrane and bending of a thin plate that describes both the experimental data and finite-elements simulations very well. We derive correction terms to account for a finite curvature of the membrane and for the inertia of the surrounding atmosphere. A simple criterion for the transition between stressed membrane and thin plate behavior is presented.

Received 14 September 2011
Publisher error corrected 25 June 2012

DOI:https://doi.org/10.1103/PhysRevB.85.035324

Corrections

25 June 2012

Erratum

Publisher’s Note: Mode shape and dispersion relation of bending waves in thin silicon membranes [Phys. Rev. B 85, 035324 (2012)]

Reimar Waitz, Stephan Nößner, Michael Hertkorn, Olivier Schecker, and Elke Scheer

Phys. Rev. B 86, 039904 (2012)

Authors & Affiliations

Reimar Waitz^*, Stephan Nößner, Michael Hertkorn, Olivier Schecker^†, and Elke Scheer

Universität Konstanz, D-78464 Konstanz, Germany

^*reimar.waitz@uni-konstanz.de
^†Robert Bosch GmBH, D-70839 Gerlingen-Schillerhöhe, Germany.

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Issue

Vol. 85, Iss. 3 — 15 January 2012

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Images

Figure 1
Sketch of the experiment showing schematically the mounting of the sample in the imaging interferometer. The interferometer is indicated by the objective lens. The sample is glued to a piezo ring used to excite the vibrations. With the help of a pump, a pressure difference is applied between upper and lower sides of the sample. It is possible to fill the chamber containing the sample with a heavy gas atmosphere as indicated by the green (light gray) shaded area.Reuse & Permissions
Figure 2
(a)–(d) Examples of measured eigenmodes (only the real part is shown) of a membrane with size 714 µm $\times$ 691 µm $\times$ 340 nm. The applied pressure difference is 50 mbar and the excitation frequency is varied from 1 to 2 MHz. Red and blue denote opposite sign of the phase with respect to the excitation. The darkness of the color is proportional to the deflection amplitude. Lower frequency modes (a) are not localized, higher modes (b)–(d) are usually located along specific paths through the membrane. For comparison with the dispersion relation we distinguish paths along the edge (b), through the center, perpendicular to the edge (d) and completely random patterns (c). (e) Dispersion relation. The black dots are experimental results from the $\vec{k}$ -space fit method, and the colored lines are calculated using Eq. (2). The colors correspond to selected paths of wave propagation with varying membrane stress (red: high stress, center of the membrane; blue: low stress, edge of the membrane; dashed green: medium stress, diagonal propagation). The apparent scattering of the experimental data is a result of the localization of modes to paths of higher or lower stress.Reuse & Permissions
Figure 3
$\vec{k}$ -space distribution ( $c_{l m}$ ) of a mode at 9.8 MHz (same membrane as in Fig. 2). The color indicates the intensity according to the scale bar at the right side. The solid lines show the fit results for the main contributions [black: peak position; red (gray): half of maximum]. The signal in the upper left corner is an artifact of a lower frequency mode.Reuse & Permissions
Figure 4
The amplitude of the $(l, m) = (3, 3)$ component of the measured modes over the excitation frequency $f$ . The $(3, 3)$ eigenfunction is shown in inset (a). The insets (b) and (c) depict the modes measured at 880 kHz and 1.008 MHz respectively. The mode (b) is a (3,3) eigenmode with only a small (1,1) contribution, resulting in the large peak. (c) is a superposition of higher modes with only a small (3,3) component.Reuse & Permissions
Figure 5
The dispersion relation of membrane bending waves (membrane size 238 µm $\times$ 269 µm $\times$ 337 nm; smaller than the one in Fig. 2). The black dots and pluses are experimental results from the maximum-amplitude method measured in air and SF $_{6}$ atmospheres respectively; the red (gray) dots are FE results. The curves show the effect of the correction terms. Unlike in Fig. 2, only curves for diagonal propagation are shown while the curves for maximum and minimum stress are omitted. The gray grid marks the wave numbers of diagonal propagation, where the data points and the shown curves should fit. The dotted blue line depicts the unperturbed result of Eq. (2). The dashed red line takes the effect of static curvature into account using Eq. (3). For the solid black line, the correction factor for air inertia [Eq. (4)] has been applied to the curved membrane dispersion.Reuse & Permissions
Figure 6
(a) The static deflection of the membrane as a function of the applied pressure difference. Experimental data are shown as well as the results of simulations using different models to take buckling into account. (b) The maximum stress in the membrane as a function of the applied pressure difference. The following FE models have been used: “FEM flat”: The membrane is modeled as a cuboid without prestress. Buckling is neglected in this case. “FEM 23.5 MPa”: A flat membrane with a compressive prestress of 23.5 MPa, leading to spontaneous buckling compensating the stress. “FEM exp. prof.”: The measured profile of the pressure free membrane is used as the equilibrium configuration of the membrane in the simulation. “FEM sinusoidal”: A sinusoidal deflection is used as equilibrium configuration of the membrane.Reuse & Permissions

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