Nonlinear stochastic dynamics of a nanomechanical resonator coupled to a single electron transistor

by Subramanian Ramakrishnan

Institution: Rutgers University
Department: Mechanical and Aerospace Engineering
Degree: PhD
Year: 2007
Keywords: Nanoelectromechanical systems; Nanoelectronics
Record ID: 1793355
Full text PDF: http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.16761


Nanoelectromechanical systems (NEMS) comprise nanometer to micrometer scale mechanical oscillators coupled to electronic devices of comparable dimensions. NEMS have great potential for sensor applications as well as for exploring fundamental physics. The dynamics of a nanomechanical resonator coupled to a single electron transistor (SET) is considered in the Duffing regime using a master equation approach and a Langevin approach. In the first approach, the master equations are derived and solved using a finite element method as well as a moment approximation method for both the single-well and the (inverted) double-well Duffing potentials. It is observed that the SET damps the resonator motion much more effectively in the single-well Duffing case in comparison with the linear case. In the double-well case we observe the existence of a limit cycle wherein the SET and the resonator exist in a state of dynamic equilibrium. This is followed by the onset of instability in the numerical solutions. The results from the master equation approach are used in a numerical fitting procedure to characterize the damping term in the averaged equations of motion of the system. It is observed that a linear damping term provides the best fit in all cases except for the strongly nonlinear regime. Based on this result, a Langevin equation is written down from which a Fokker-Planck equation is derived for the system. The Fokker-Planck equation is solved analytically, in closed form, for the steady state. In the time dependent case, the equation is solved using a finite element method and the results are shown to be in qualitative agreement with those obtained using the master equation approach. Therefore it is established that the SET-resonator system attains a steady state much more rapidly in the single-well Duffing regime. Finally, the steady state analytical solution to the Fokker-Planck equation is utilized to show that the steady state effective temperature is lower in the presence of the single-well Duffing nonlinearity.