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

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-s hift inlerferometer lI sin g strobosco pic li ght. The observed wave patlelll of th e 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 thi s 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 ht:ndin g or a thin plale lhat dt:s~rihes both tht: t:xperimel1la l data and finite-dt:mt:nts simulations vt:ry wdl. We dt:rivt: correc li on term s to account for a finil e curvalure or th e membrane and for the il1l:nia of th e surrounding atmosphere. A simple criterion for the transition between stressed membrane and thin plate behavior is presented.