Charge transport through a single-electron transistor with a mechanically oscillating island

We consider a single-electron transistor (SET) whose central island is a nanomechanical oscillator. The gate capacitance of the SET depends on the mechanical displacement, thus, the vibrations of the island may influence the transport properties. Harmonic oscillations of the island and thermal vibrations change the transport characteristics in different ways. The changes in the Coulomb blockade oscillations and in the current noise spectral density help to determine in what way the island oscillates, and allow to estimate the amplitude and the frequency of the oscillations.

Figure 1

(a) Sketch of a single-electron transistor (SET) with an oscillating island. The island is coupled to the left (L) and right (R) leads by tunnel junctions, and its capacitance to the gate $C_g\left(z\right)$ depends on the coordinate $z$ that measures the deviation of the island from its equilibrium position. (b) Equivalent circuit of the device.

Figure 2

(Color online) Current gate-charge characteristics for a symmetric SET $\left(R_{\mathrm{L}}=R_{\mathrm{R}}\right)$ with $C_g⪢C_{\mathrm{L},\mathrm{R}}$, $V_{\mathrm{L}}=-V_{\mathrm{R}}=V∕2$, $Q_0=-C_g^{\left(0\right)}{V}_g$. (a) $V=0.2$, $\left(\mid V\mid ∕2<V_{\mathrm{osc}}\right)$, (b) $V=0.5$, $\left(\mid V\mid ∕2\approx V_{\mathrm{osc}}\right)$, (c) $V=1$, $\left(\mid V\mid ∕2>V_{\mathrm{osc}}\right)$. The voltage is measured in units of $\mid e\mid ∕2C_{\Sigma }(z=0)$, the current in units of $I_0=(e∕C_g^{\left(0\right)})∕(R_{\mathrm{L}}+R_{\mathrm{R}})$. The dashed curve corresponds to a static island, the solid curve to a harmonically oscillating island, $z=z_0\sin \left({\omega }_{0}t\right)$; $z_0\left({\partial }_z{C}_g\right)∕C_g=5.6\times {10}^{-3}$ (this is typical for SETs where the island is a nanotube;[10] then $z_0\approx 5r$, where $r$ is a typical nanotube diameter). The dotted curve in Fig. 1a illustrates what happens if $V_{\mathrm{osc}}∕(\mid e\mid ∕2C_g^{\left(0\right)})=5>1$ and Eq. (8) is not valid. The dash-dotted curves correspond to the case of thermal motion; the thermal average ${⟨z^2⟩}_T\equiv k_{B}T∕m{\omega }_0^2$ is chosen to be equal ${⟨{\left[z_0\sin \left({\omega }_{0}t\right)\right]}^2⟩}_t=z_0^2∕2$. The integer part of $Q_0∕e$ is the number of electrons on the island in the static regime when $V_{\mathrm{L}}=V_{\mathrm{R}}=0$. The curves are periodic in the static case, but not if the island oscillates. The areas under the peaks in the static and in the dynamical cases are the same.

Figure 3

(Color online) Fano factor for $V<V_{\mathrm{osc}}$. The dashed line corresponds to the static case, the solid line to the harmonically oscillating island, and the dash-dotted line to the thermally driven island. The average square amplitude of the island oscillations are equal for the harmonically oscillating island and the case of thermal equilibrium.