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

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.

Non-linear vibrational response of Ge and SiC membranes

“…8 Resonant vibration measurements are also starting to see more applications, and involve vibrating a membrane and using a laser interferometer or vibrometer to study its resonant modes. 1,[12][13][14][15][16][17] Previous measurements and calculations have predominantly measured small amplitude membrane vibrations, where the vibrations can be approximated as being in the linear regime. 1,12,18,19 However, if the vibration amplitude is greater than a critical value, bending and stretching during vibration becomes important.…”

“…where h is the membrane thickness, q is the density, z the out-of-plane deflection, and r ij the stress tensor of the membrane. 1,15,25 For small deflection amplitudes the stress is constant, given by the static pre-stress, and therefore, the motion is approximately linear. The resonant frequencies of a rectangular membrane in this regime (without considering air damping) are given by…”

Spatially Resolved Measurement of the Stress Tensor in Thin Membranes Using Bending Waves

“…Both types are a few hundred nanometers thick and have a lateral size of roughly 500 μm. They are fabricated from coated silicon wafers using a wet-etching process [8] adapted from Ref. [[12]].…”

“…A small static buckling is caused by the formation of native oxide leading to a compressive stress [8,14,15]. During the measurement, we apply a pressure difference of 50 mbar between the top and the bottom side of the membrane, so it is bent by 8 μm.…”

“…In a previous work [8], we show how to extract the dispersion relation of bending waves of prestressed thin plates from measured mode shapes. Stress and bending stiffness of the membrane are determined as fit parameters using a theoretical model of the dispersion relation.…”