Dynamics of a Duffing nanomechanical resonator coupled to a single-electron transistor: A master equation approach

Abstract

We investigate the nonlinear dynamics of a classical Duffing nanomechanical resonator coupled to a single-electron transistor (SET) using a master equation formalism. We consider both the cases of hardening and softening stiffnesses of the Duffing regime while assuming linear coupling between the SET and the resonator. We first derive scaled master equations for the coupled system and define a parameter that characterizes the effective nonlinearity in the system. Solving the equations using a method of moment approximations and validating the approximations independently using finite element solutions, we conclude that the coupled system reaches a steady state and that interaction with the SET damps the motion of the resonator at a significantly higher rate in the hardening Duffing case in comparison with the case of a linear harmonic resonator. The concomitant conclusion that the steady state is attained more rapidly suggests that the hardening stiffness Duffing regime has better prospects for sensing applications than the linear regime. Moreover, analysis of the variance of the resonator displacement in the steady state indicates that lower steady-state effective temperatures are obtained in the nonlinear case. In the case of softening stiffness, dynamical instability occurs in certain parameter regimes implying that the SET transfers energy to the resonator in these regimes. Interestingly, the onset of instability is preceded by the appearance of periodic orbits in phase space. Since weak coupling between the SET and the resonator is assumed throughout, our analysis indicates that a variety of important phenomena, including negative damping, can arise from purely nonlinear motion of the nanomechanical resonator in this regime.