Double coupled electron shuttle

A nanoshuttle consisting of two movable islands connected in series and integrated between two contacts is studied. We evaluate the electron transport through the system in the presence of a source-drain voltage with and without an rf excitation. We evaluate the response of the system in terms of the net direct current enhanced by the mechanical motion of the oscillators. An introduction to the charge stability diagram is given in terms of electrochemical potentials and mechanical displacements. The low capacitance of the islands allows the observation of Coulomb blockade even at room temperature. Using radio frequency excitations, the nonlinear dynamics of the system is studied. The oscillators can be tuned to unstable regions where mechanically assisted transfer of electrons can further increase the amplitude of motion, resulting of a net energy being pumped into the system. The resulting amplified response can be exploited to design a mechanical motion detector of nanoscale objects.

Figure 1

(a) SEM image of a coupled shuttle consisting of two Si-based nanopillars, (b) double harmonic oscillator, with coupling spring constants $k$ and $k_0$, and mechanical displacements $x_L$ and $x_R$, and (c) forces that generate the displacement $x$ in a differential element $ds$ of the nanopillar. (d) The flexural mode where the center of mass is at rest.

Figure 2

A schematic picture of the double-island structure with the voltage sources and various capacities in the system. The two circles denote the islands with $n_{L,R}$ excess electrons. The distance between the islands is given in terms of their relative displacement, $x\left(t\right)$. A bias $V\left(t\right)$ is applied to the left contact while maintaining the right one grounded.

Figure 3

Schematic representation of the tunneling processes in a system of two islands attached to the S and D leads. Six different processes are given in terms of chemical potentials, ${\mu }_i^{\rightleftarrows }$, $i$ $=$ 1, 2, 3, as indicated in the picture.

Figure 4

Contour plot of the absolute value of the direct current $|I_{\mathrm{dc}}|$ as a function of $V_S$ and $V_G$. The radii of the nanoislands were set to $r_L=28$ nm and $r_R=31$ nm. Coulomb blockade diamonds are apparent, even at room temperature.

Figure 5

Electronic transport through a double pendula structure. The direct current is to the right (a) [left, (b)] direction if the metallic grains approach the contacts (each other) when the applied bias is negative. Electron transfer between the islands to the right (left) direction occurs in (out of) phase with the signal, as depicted in the bottom of (a) [top of (b)].

Figure 6

Schematic representation of the $p$ $=$ 1 tongue in the parameter space spanned by ${\delta }_{\omega }$ and ${\alpha }_1$. In between the transition curves (broken lines), the trivial solution is unstable and two stable solutions are born, $\varphi$ $=$ 0, $\pi$ (solid curves). The insets depict phase portraits in the $A$-$B$ plane: below the lower transition curve, only the trivial solution exists. Between both transition curves, the trivial solution is unstable, and another two solutions are born. Above the second transition curve, $r_0$ $=$ 0 is stable again and the other two solutions become unstable.

Figure 7

Contour plot of the direct current $I_{\mathrm{dc}}$ as a function of ${\alpha }_1$ and ${\delta }_{\omega }$ in the first tongue, $p=1$. The inset shows a phase portrait in the $A$-$B$ plane: Inside the tongue, the origin is unstable, and two stable solutions are found, $P_{\pi /2}$ and $P_{3\pi /2}$.

Figure 8

Time evolution of $n\left(t\right)=n_l-n_R$ in the stationary limit for low (thin solid curve) and high bias (thick solid curve). In the high-bias regime, $n\left(t\right)$ can be approximated by a square wave.

Figure 9

A schematic picture of the double-island structure with the voltage sources and various capacities in the system. The three junctions are characterized by $m_j$ and $V_j$, $j$ $=$ 1, 2, 3. The islands are coupled to each other, with a mutual capacitance $C_2$, as well as to a gate voltage, $V_G$, with capacitances $C_{G1}$ and $C_{G2}$, respectively. Also, they are coupled to the source-drain leads, with capacitances $C_1$ and $C_3$.