Current blockade in classical single-electron nanomechanical resonator

We consider a single-electron transistor where the central metallic island can oscillate. It has been shown that for a weak coupling of the elastic and electric degrees of freedom, the position of the island fluctuates with a small variation of the current through the device. In this paper, we consider the strong coupling limit. We show that the system undergoes a static mechanical instability that is responsible for the opening of a gap in the current-voltage characteristics even at the degeneracy point. We provide an analytical description of the transition point, taking into account the nonequilibrium mechanical state. We also discuss how the mechanical nature of the suppression of the current can be probed experimentally by a slow modulation of the gate voltage.

Figure 1

Left panel: Circuit diagram for the single-electron transistor with an oscillating central island. The variable $X$ parametrizes an effective position. Right panel: Conducting regions shaded in the $V_g-V$ near a degeneracy point between $N$ and $N+1$ electrons.

Figure 2

(Color online) Left panel: Effective potential $U_{\mathit{eff}}$ for $v_g=-1∕2$ and $v=1.05$, 1, and 0.95. Right panel: Shape of the conducting regions (filled regions) in the plane $v_g-v$ when the displacement of the mobile part is taken into account. The dashed lines indicate the border of the Coulomb diamonds for $x=0$ and $x=-1$. The small shaded diamond indicates the mechanically bistable region.

Figure 3

(Color online) Current as a function of the reduced gate $\left(v_g\right)$ and bias voltages $\left(v\right)$ in the mean field approximation of Eq. (7) with $x=x_m$. The opening of a gap is clearly visible together with the fact that the current is continuous at the boundary of the conducting region. The border of the Coulomb diamonds for $x=0$ is shown dashed.

Figure 4

Lines of equal energy and, thus, of equal probability $\mathcal{Q}$ in the $x-u$ plane for the values of $v$ and $v_g$ indicated.

Figure 5

(Color online) Current as a function of $v_g$ and $v$, including the effect of position fluctuation. The dependence on $v_g$ is reduced inside the conducting region with respect to the mean field result due to the average over the different positions visited by the oscillating part.

Figure 6

Comparison of the contour plots of the current in mean field approximation (left) and taking into account the fluctuations (right) as a function of the gate $\left(v_g\right)$ and bias voltage $\left(v\right)$.

Figure 7

Probability distribution for the position at the apex of the conducting region.

Figure 8

(Color online) Current as a function of $v$ at $v_g=-1∕2$ from the analytical calculation (solid line), the weak coupling theory (dashed line), and the Monte Carlo simulation (points). The values of ${\Gamma }_o$ are 10 (empty square), 1 (triangle), 0.1 (cross), and 0.01 (filled square). Inset: Current as a function of $v_g$ for $v=1.5$. The long dashed line is Eq. (7) for $x=x_m$.

Figure 9

(Color online) Average current response to an oscillating gate voltage $v_g=-1∕2+v_g^o\sin \left(\omega t\right)$ as a function of $v_g^o$. $v=1∕2$ and $\omega ∕{\omega }_o=0.1$ (same symbols as in Fig. 8). The arrow indicates the threshold to the conducting region. Inset: Frequency dependence of the average current for $v_g^o=1.05$ the threshold value. Structures at $\omega ∕{\omega }_o=0.5$, 1, and 2 are clearly visible.