Nanoscale mechanical resonators and oscillators

In this thesis the physics of nanoscale mechanical resonators and oscillators is studied. We discuss two types of resonators. First, a top-down fabricated doubly clamped beam resonator with an integrated piezoelectric actuator is introduced. The second type of resonators are based on layered two-dimensional materials, such as graphene and molybdenum disulphide (MoS2). In chapter 2 we discuss the dynamics of doubly clamped microbeam resonators. These resonators have an integrated piezoelectric actuator. To enhance the Q factor of these devices we employ an external feedback loop, and operate the resonator as an oscillator. With the feedback loop in place, we show a twenty times enhancement of the effective Q-factor. When the gain in the feedback loop exceeds the oscillation threshold, the oscillator becomes nonlinear and the frequency now depends on the gain and phase of the feedback loop. Two types of transitions occur: one showing a continuous transition from the linear to the nonlinear regime, and one displaying an instable transition with hysteresis. Next to top-down mechanical resonators, we have also considered resonators based on layered two-dimensional materials. To reliable measure the motion of these resonators, which is challenging due to the limited thickness - of in the extreme case a single layer - we developed a Fabry Perot interferometer. The interference occurs between light reflected from two mirrors placed after each other. In this case, the front mirror is a suspended drum made from a layered material and the back mirror is the silicon substrate. Due to the motion of the drum the cavity length changes in time which results in a modulation of the reflected light intensity. To optimize the resonator geometry we derived an equation for the cavity reflectivity as a function of device parameters. In the first experiment done with the interferometer we measured the properties of suspended MoS2 drum resonators with thicknesses ranging from a single layer to several layers. By measuring the frequency spectra of multiple drums we showed that resonators made of one to five layers exhibit membrane-like behavior. Drum resonators made of more than five layers display plate-like behavior. We further notice that the Q-factor measured at room temperature is remarkable low compared to other nanomechanical systems and is of the order of hundred. To investigate the Q-factor of drum resonators at room temperature we developed a measurement technique to determine the energy relaxation of the resonator in the time domain. This has been done by mixing the measured signal coming from the photodiode with the drive frequency. The mixing yields a component at dc and a component at approximately two times the drive frequency. By recording the dc component on an oscilloscope the ring up and ring down has been measured. By comparing the Q-factor from the measurement in the frequency domain with the Q-factor measured in the time domain, one can determine if the Q-factor is limited by dissipation or by spectral broadening. We found that the Q-factors in both measurements are equal. This indicates that in our drum resonators dissipation is the dominant loss factor at room temperature. In the last chapter we study the dynamics of a graphene oscillator with an external feedback loop. The feedback loop consists of a voltage controlled phase shifter and a variable gain amplifier. By adjusting the phase in the feedback loop we can access multiple regimes. First, we explore the regime of positive velocity proportional feedback, the same as we have done in chapter 2. We show a thirty five times enhancement of the effective Q-factor by applying feedback. By choosing the opposite phase, the damping is increased; this leads to a cooling of the oscillator motion and a smaller peak in the spectral response. As also observed in chapter 2, the oscillator becomes nonlinear and the oscillation frequency depends on the loop gain and phase when the gain exceeds the threshold for oscillation