Frequency Noise and Self-sustained Oscillations in Microelectromechanical Systems

Abstract

We experimentally characterize fluctuations in the eigenfrequency of a resonantly driven micromechanical resonator when the frequency noise is of the telegraph type. The time-averaged vibration amplitude spectrum of the resonator exhibits two peaks. They merge with an increasing rate of frequency switching and the spectrum displays an analog of motional narrowing. We show that the scaled higher moments of the complex vibration amplitude depend strongly on the frequency noise characteristics. This dependence remains valid even when strong detector noise is present. We also study self-sustained oscillations of electromechanical resonators with different mechanisms. In the first case, self-sustained oscillations take place due to negative damping induced by the dynamical backaction in a micromechanical resonator that is designed to have two nonlinearly coupled vibrational modes with strongly differing frequencies and decay rates. We find that self-sustained oscillations are induced in both modes. The nonlinear nature of the backaction leads to hysteresis of the vibrations. In each mode, the vibration phase undergoes anomalous diffusion, where the time dependence of the phase variance follows a superlinear power law. We show that the exponent of the power law is determined by the exponent of the 1/f-type intrinsic frequency noise of the resonator. The phase fluctuations of the two modes show near perfect anticorrelation. The other self-oscillations are realized by coupling a conventional phase locked loop to a doubly clamped beam resonator whose motion is detected by the magnetomotive reflection technique. This technique usually suffers from large signal background due to impedance mismatch of the electrical resistance $R_e$ of resonators to the impedance ($50 \ \Omega$) of the detection cables. We minimize the signal background by patterning a gold nanowire on the top of the beam and controlling $R_e$ of the nanowire via electromigration. We show that self-sustained oscillations of the beam are successfully excited as $R_e$ is increased near $50 \  \Omega$