Diffusion-Induced Nonlinear Dynamics in Carbon Nanomechanical Resonators
Licentiate thesis, 2015
The emergence of nanoelectromechanical systems has enabled the development of sensors capable of detecting mass, charge, force, position, and spin with an unprecedented precision. In particular, the low mass, high resonant frequency, and high quality factor of carbon nanomechanical resonators make them ideal for the creation of a high sensitivity mass sensor. Carbon nanotube resonators are indeed the basis of the most sensitive mass sensors to date, whereas resonators made from suspended graphene monolayers are potentially capable of a very high rate of operation, due to their large surface area. A complicating factor is that the available mass measurement schemes rely on that the measured mass remains stationary, something that is no longer true at non-cryogenic temperatures.
In this thesis, the effect of an elevated ambient temperature on a mass-resonator system is studied by simulating ring-down experiments. Thermal fluctuations in the position of the mass on the resonator introduce a stochastic force in the system equations of motion; these stochastic differential equations are here solved analytically and numerically. The unperturbed resonator is modeled as an undamped linear oscillator, but the addition of a diffusing mass induces nonlinear dynamics in the system. The presence of the particle mediates a coupling between vibrational modes, that acts as a new dissipation channel. Additionally, short-time correlations between the motion of the diffusing particle and the vibrating resonator results in a second dissipation mechanism, that causes a nonexponential decay of the vibrational energy. For vibrational amplitudes that are much larger than the thermal energy this dissipation is linear; for small amplitudes the decay takes the same form as that of a nonlinearly damped oscillator.
stochastic differential equations
equations of motion