Charge Shuttle as a Nanomechanical Rectifier

We consider the charge shuttle proposed by Gorelik et al. driven by a time-dependent voltage bias. In the case of asymmetric setup, the system behaves as a rectifier. For pure ac drive, the rectified current shows a rather rich frequency dependent response characterized by frequency locking at fractional values of the external frequency. Because of the nonlinear dynamics of the shuttle, rectification is present also for very low frequencies. These effects could be useful to unveil the internal dynamics of nanomechanical devices.

Figure 1

Left panel: Schematic of a charge shuttle. Right panel: Energy diagram for the SET. The diagonal lines indicate the thresholds for the vanishing of the four rates ${\Gamma }_{FL}$, ${\Gamma }_{FR}$, ${\Gamma }_{TL}$, and ${\Gamma }_{TR}$. The two dots indicate the state of the system during shuttling at fixed voltage $V$.

Figure 2

Current as a function of the frequency for $ϵ=0.5$, $\gamma /{\omega }_o=0.05$, $\Gamma /{\omega }_o=1$, and $R_R/R_L=10$. The result of the simulation of the stochastic dynamics (grey or green points) is compared with the approximate $I_a$ (solid line). In the small frequency region, enlarged in the upper inset, several resonances at fractional values of ${\omega }_o$ appear. We also show (horizontal dark grey or red line) the analytical result from Eq. (4) in the adiabatic limit. The triangular dot indicates the numerical solution of the adiabatic equations. Lower inset: Current noise from the simulation (grey or green points) and analytic result (dashed line) for the static SET.

Figure 3

The top panel shows the calculated winding number obtained by the stochastic simulation (green points) and by the average approximation of Eq. (2) (black points). The bottom panel shows $\Delta \varphi$ obtained with different methods: (i) average approximation (solid line), (ii) stochastic simulation with $w$ from Eq. (5) (green points), (iii) stochastic simulation with $w$ set equal to 1 (red dashed line), or (iv) with $w=2$ (blue dashed line). Parameters are the same as in Fig. 2.