Brownian Fluctuations of a Nanomechanical String Resonator Immersed in a Viscous Fluid

The precision of a measurement using a mechanical resonator is limited by Brownian motion, especially for a small resonator such as a nanoelectromechanical systems (NEMS) resonator. Here, we investigate the Brownian dynamics of a doubly-clamped nanomechanical beam under high tension immersed in a viscous fluid both theoretically and experimentally. Our beam is modeled as an oscillating cylinder so that Stokes theory can be applied to obtain an analytic form for fluid dissipation. Due to the high tension, the beam obeys the equation of motion of a fluid-loaded string, from which we find its mechanical susceptibility as a function of frequency and position along the beam. The position-dependent power spectral density (PSD) of the thermal displacement fluctuations of the beam is then obtained from the susceptibility via the fluctuation-dissipation theorem. In order to verify our theory, we use optical interferometry to measure the displacement fluctuations of a 50 μm × 900 nm × 100 nm silicon nitride beam in air and water. The PSD of the beam's displacement noise in air shows distinct peaks at the eigenfrequencies, which we integrate over frequency to obtain the modal spring constants. The experimentally-determined eigenfrequencies and spring constants are then inserted into our analytically-derived expression to predict the PSD of displacement fluctuations in water. We obtain excellent agreement between theory and experiments in water for different positions along the beam, observing broad and overlapping modal peaks.