Electrical transport through a single-electron transistor strongly coupled to an oscillator

We investigate electrical transport through a single-electron transistor coupled to a nanomechanical oscillator. Using a combination of a master-equation approach and a numerical Monte Carlo method, we calculate the average current and the current noise in the strong-coupling regime, studying deviations from previously derived analytic results valid in the limit of weak coupling. After generalizing the weak-coupling theory to enable the calculation of higher cumulants of the current, we use our numerical approach to study how the third cumulant is affected in the strong-coupling regime. In this case, we find an interesting crossover between a weak-coupling transport regime where the third cumulant heavily depends on the frequency of the oscillator to one where it becomes practically independent of this parameter. Finally, we study the spectrum of the transport noise and show that the two peaks found in the weak-coupling limit merge on increasing the coupling strength. Our calculation of the frequency dependence of the noise also allows one to describe how transport-induced damping of the mechanical oscillations is affected in the strong-coupling regime.

Figure 1

(Color online) Circuit diagram of the system studied. The gate capacitance of the SET depends on the displacement of a mechanical oscillator, leading to a coupling of the electrical transport through the device and the mechanical motion of the oscillator.

Figure 2

Upper panel: total probability distribution $P\left(x\right)=P_N\left(x\right)+P_{N+1}\left(x\right)$ of the oscillator’s position for different values of the coupling constant $\kappa$ [defined in Eq. (7)]. Lower panel: probability distribution $P_N\left(x\right)$ to find the oscillator at position $x$ if the SET is in charge state $N$. $P_{N+1}\left(x\right)$ can be obtained by the symmetry relation $P_{N+1}\left(x\right)=P_N(1∕2-x)$. In both panels, lines are shifted for clarity by $2\kappa$ and the difference between neighboring curves is $\Delta \kappa =0.05$. All calculations were done at $ϵ=0.3$ and at the charge-degeneracy point. For the definition of $ϵ$, see Eq. (4).

Figure 3

(Color online) Two-dimensional phase-space distributions $P(x,u)=P_N(x,u)+P_{N+1}(x,u)$, for $\kappa =0.2$ (left panel) and $\kappa =0.9$ (right panel).

Figure 4

(Color online) Current (in units of $e∕{\tau }_t$) as a function of $\kappa$ at the degeneracy point $⟨P_N⟩=⟨P_{N+1}⟩$, for $ϵ=0.3$. The dots are the results of the Monte Carlo calculation, and the solid line is the analytic form found within the weak-coupling approximation.

Figure 5

(Color online) Normalized third cumulant as a function of $\kappa$ for different values of $ϵ$, as calculated within the weak-coupling approximation and scaled by ${ϵ}^4$. Dotted line: $ϵ=0.4$. Dashed line: $ϵ=0.3$. Solid line: $ϵ=0.2$. Dash-dotted line: $ϵ=0.1$. The inset shows the $ϵ$ dependence of the normalized third cumulant at $\kappa =0.1$ (symbols), as calculated within the weak-coupling approximation. The solid line is a fit to a power law $\sim {ϵ}^{-4}$. These results were obtained by integrating the equation of motion for $⟨n^3\left(t\right)⟩$ following from Eq. (13).

Figure 6

(Color online) Upper panel: Fano factor as a function of $\kappa$ at the degeneracy point $⟨P_N⟩=⟨P_{N+1}⟩$. The dots are the results of the numerical calculation, and the solid line is the analytic form found within the weak-coupling approximation. Lower panel: normalized third cumulant ${\mu }_3$ of the probability distribution ${\mathcal{P}}_n$. For both panels, $ϵ=0.3$.

Figure 7

(Color online) Fano factor (upper panel) and normalized third cumulant (lower panel and inset) as a function of $\kappa$ for different values of $ϵ$: $ϵ=0.1$ (squares in the upper panel and in the inset), $ϵ=0.2$ (diamonds), $ϵ=0.3$ (circles), and $ϵ=0.4$ (triangles). Note the logarithmic $y$ axis in the upper panel.

Figure 8

(Color online) Frequency-dependent noise power beyond the weak-coupling approximation. For each curve, the SET is tuned to the charge-degeneracy point and $ϵ=0.3$.

Figure 9

(Color online) Position ${\omega }_0^{\prime }$ of the first peak in the frequency-dependent noise power as a function of $\kappa$. The solid line gives the weak-coupling prediction ${\omega }_0\sqrt{1-\kappa }$, the data points are the numerical Monte Carlo results, and the dashed line was obtained using an effective damping constant; see text.