Nonlinear dynamics of a driven nanomechanical single-electron transistor

We analyze the response of a nanomechanical resonator to an external drive when it is also coupled to a single-electron transistor (SET). The interaction between the SET electrons and the mechanical resonator depends on the amplitude of the mechanical motion leading to a strongly nonlinear response to the drive which is similar to that of a Duffing oscillator. We show that the average dynamics of the resonator is well described by a simple effective model which incorporates damping and frequency renormalization terms which are amplitude dependent. We also find that for a certain range of parameters the system displays interesting bistable dynamics in which noise arising from charge fluctuations causes the resonator to switch slowly between different dynamical states.

Figure 1

Circuit diagram of the SET resonator system. The SET is coupled by tunnel junctions to two leads. A gate capacitor is used to tune the operating point, one plate of which is mechanically compliant to provide electromechanical coupling.

Figure 2

Oscillations in both the mean and variance of the position distribution over a period of the drive. The parameters are $\epsilon =0.3$, $\kappa =0.05$, $f_0=0.5$, and ${\omega }_D=0.002\pi$. In (a) we show the full position distribution (obtained from a Monte Carlo calculation) and the analytic solution for the mean as the dashed (white) line. In (b) we show the variance; the solid (black) line is the numerical result, and the dashed (black) line shows the solution to Eqs. (28, 29, 30, 31, 32, 33, 34). For comparison we also show as the dash-dotted (red) line the variance expected for an adiabatically slow force, from Eq. (35), and the horizontal line shows the variance corresponding to the average temperature experienced over the cycle. Note that for all numerical calculations we chose ${\Delta }_L,{\Delta }_R$ so that the system is at the charge degeneracy point, $\langle P_1\rangle =1/2$, where the oscillations in $\langle P_1\rangle \langle P_0\rangle$ at frequency ${\omega }_D$ vanish.

Figure 3

Amplitude-dependent (a) damping and (b) frequency of the resonator. Solid (black) lines show the solution using Eq. (4b); dashed (red) lines show the results using Eq. (39). The parameters used are $\epsilon =0.3$, $\kappa =0.05$, $f_0=0.015$, and ${\omega }_D=0.29$.

Figure 4

The solid (black) lines show the stable solutions to the self-consistency expression, Eq. (40), as a function of drive frequency, for $f_0=0.02$, $\epsilon =0.3$, and $\kappa =0.05$. We also show the linear result as the dashed (black) curve and the amplitude at which the linear theory fails, $A_c$, as the horizontal dot-dashed (blue) line. The shaded regions (a) and (b) are discussed in the text.

Figure 5

Amplitude of the resonator (a) and average current (b) as a function of drive frequency. Results from a Monte Carlo simulation with a forward [solid (red) line] and backward [dot-dashed (green) curve] frequency sweep for $f_0=0.025$ are shown together with the fully linear results for $f_0=0.01$ [solid (black) curve]. Notice that for each sweep the initial conditions for a given frequency were chosen to match the final state found for the previous value of the frequency. The results obtained using amplitude-dependent effective damping and frequency shift [Eq. (50)] are also shown in (a) plotted as a dashed (blue) curve. The other parameters used are $\epsilon =0.3$ and $\kappa =0.05$.

Figure 6

Variance of the $x$ distribution averaged over an integer number of drive periods, $\langle \delta {\tilde{x}}^2\rangle$, for various drive strengths; the solid (black) curve is $f_0=0.01$ (linear), the dashed (purple) curve is $f_0=0.015$ (bistable), and the dot-dashed (green) curve is $f_0=0.02$ (nonlinear). These results were obtained from Monte Carlo simulations in which the frequency was swept downwards.

Figure 7

Response of the system as a function of driving frequency; all parameters are the same as in Fig. 4, except $f_0=0.015$. The solid line shows the numerical response, and the dashed line shows the analytically predicted stable solutions using the damping and frequency shift from Eq. (50). The system shows a bistability in the region where the linear solution reappears.

Figure 8

Example trajectories of the amplitude and phase of the resonator along with the current through the SET as a function of time. The dashed lines show the point at which we choose to split the data into the high- and low-amplitude states. The parameters are $\epsilon =0.3$, $\kappa =0.05$, and $f_0=0.016$. The drive frequency is chosen such that the occupation probabilities of the two states are equal. Each point in the plots is averaged over 200 periods of the drive.

Figure 9

(a) Transition rates between the two states as a function of drive frequency, close to the bistability. The (blue) dots denote ${\Gamma }_{hl}$, the rate from the high-amplitude state to the low-amplitude state, and the (red) crosses show ${\Gamma }_{lh}$. The insets show the resonator probability distributions at the drive frequencies labeled (i), (ii), and (iii) in the main panel. The parameters at point (ii) are the same as in Fig. 8. (b) Corresponding zero-frequency Fano factor, calculated from the two-state model, as described in the text. The points labeled (i), (ii), and (iii) match those in (a).