Large current noise in nanoelectromechanical systems close to continuous mechanical instabilities

We investigate the current noise of nanoelectromechanical systems close to a continuous mechanical instability. In the vicinity of the latter, the vibrational frequency of the nanomechanical system vanishes, rendering the system very sensitive to charge fluctuations and, hence, resulting in very large (super-Poissonian) current noise. Specifically, we consider a suspended single-electron transistor close to the Euler buckling instability. We show that such a system exhibits an exponential enhancement of the current noise when approaching the Euler instability which we explain in terms of telegraph noise.

Figure 1

Sketch of a suspended doubly clamped nanobeam forming a quantum dot (solid black line) connected via tunnel barriers to source and drain electrodes held at chemical potentials ${\mu }_{\mathrm{L}}$ and ${\mu }_{\mathrm{R}}$ by the bias voltage $V$. The lateral force $F$ compresses the nanobeam and induces the buckling instability. The beam is capacitively coupled to a metallic electrode kept at a gate voltage $V_{\mathrm{g}}$. This induces a stochastic force $F_{\mathrm{e}}$ that attracts the beam towards the gate electrode whenever the quantum dot is charged (dashed red line), inducing fluctuations of the nanobeam's deflection and, in turn, fluctuations of the current through the device.

Figure 2

Zero-temperature (a),(d) average occupation of the dot for fixed $x$ [Eq. (7)], (b),(e) quasistationary current [Eq. (14)] and (c),(f) effective potential [Eq. (15)], (a)–(c) far below the Euler instability ($-\delta \gg {\tilde{\alpha }}^{1/3}$) and (d)–(f) at the instability ($\delta =0$) for increasing bias voltages and for a gate voltage $v_{\mathrm{g}}=v_{\mathrm{g}}^{\min }$ [cf. Eq. (17)].

Figure 3

For increasing temperature, (a),(d) average occupation of the dot for fixed $x$, (b),(e) quasistationary current, and (c),(f) effective potential, (a)–(c) far below the Euler instability ($\delta =-1$), and (d)–(f) at the instability ($\delta =0$). In the figure, $v={\Delta }_v$, $v_{\mathrm{g}}=v_{\mathrm{g}}^{\min }$, and $\tilde{\alpha }={10}^{-6}$.

Figure 4

Fano factor as a function of the reduced compression force $\delta$ [cf. Eq. (2)]. In the figure, ${\gamma }_{\mathrm{e}}={\omega }_0/\Gamma ={10}^{-2}$, $\tilde{T}=3$, $\tilde{\alpha }={10}^{-6}$, $v={\Delta }_v$ and $v_{\mathrm{g}}=v_{\mathrm{g}}^{\min }$. The red circles correspond to the data points, while the dotted line serves as a guide to the eye.

Figure 5

Deflection $x$ (in blue) and quasistationary current $\mathcal{I}$ (in red) as a function of time simulated by the Langevin equation (5). In the figure, the compression force increases from (a) to (e): (a) $\delta =-1$, (b) $\delta =-0.05$, (c) $\delta =0$, (d) $\delta =0.05$, and (e) $\delta =1$. The parameters used in the figure are the same as in Fig. 4.

Figure 6

(a) Current (23) and (b) zero-frequency noise (24) as a function of bias voltage for increasing temperature as obtained from the telegraph-noise model.

Figure 7

(a) Current and (b) zero-frequency noise as a function of bias voltage for increasing values of the reduced compression force $\delta$. In the figure, $v_{\mathrm{g}}=v_{\mathrm{g}}^{\min }$, $\tilde{T}={\Delta }_v/10$, ${\gamma }_{\mathrm{e}}={10}^{-2}$, ${\omega }_0/\Gamma =0$ (no current-induced fluctuations), and $\tilde{\alpha }={10}^{-6}$ for all lines except for the thick blue dashed one, where $\tilde{\alpha }={10}^{-3}$.

Figure 8

(a) Current and (b) zero-frequency noise as a function of bias voltage for increasing temperature $\tilde{T}=0.03$ (solid line), $0.04$ (dashed line), $0.06$ (dotted line), $0.08$ (dash-dotted line). In the figure, $\delta =-1$, $v_{\mathrm{g}}=v_{\mathrm{g}}^{\min }$, ${\gamma }_{\mathrm{e}}={10}^{-2}$, ${\omega }_0/\Gamma =0$, and $\tilde{\alpha }={10}^{-6}$. Insets: Same as the main figure for $\delta =0$. In the insets, $\tilde{T}=2$ (solid line), $2.5$ (dashed line), 3 (dotted line), $3.5$ (dash-dotted line).

Figure 9

Zero-frequency noise as a function of bias voltage for increasing values of the extrinsic damping constant (or inverse quality factor) ${\gamma }_{\mathrm{e}}$. In the figure, $\delta =-1$, $v_{\mathrm{g}}=v_{\mathrm{g}}^{\min }$, $\tilde{T}={\Delta }_v/10$, ${\omega }_0/\Gamma =0$, and $\tilde{\alpha }={10}^{-6}$. Inset: Same as the main figure for $\delta =0$.

Figure 10

Frequency dependence of the mechanical noise for various values of the bias voltage. In the figure, $\delta =-1$, $v_{\mathrm{g}}=v_{\mathrm{g}}^{\min }$, $\tilde{T}={\Delta }_v/10$, $\tilde{\alpha }={10}^{-6}$, ${\gamma }_{\mathrm{e}}={10}^{-2}$, and ${\omega }_0/\Gamma =0$. Inset: Same as the main figure for $\delta =0$.

Figure 11

(a) Current and (b) zero-frequency noise as a function of bias voltage for increasing values of the adiabaticity parameter ${\omega }_0/\Gamma$. In the figure, $\delta =0$, $v_{\mathrm{g}}=v_{\mathrm{g}}^{\min }$, ${\gamma }_{\mathrm{e}}={10}^{-2}$, $\tilde{T}/{\Delta }_v=0.1$ and $\tilde{\alpha }={10}^{-6}$. Inset: Same as the main figure with $\delta =-1$.

Figure 12

Zero-frequency shot noise as a function of bias $v$ and gate voltage $v_{\mathrm{g}}$ (a) for vanishing compression force $F=0$ and (b) at the Euler buckling instability where $F=F_{\mathrm{c}}$. The parameters used in the figure are $\tilde{T}=10$, $\tilde{\alpha }={10}^{-6}$, ${\omega }_0/\Gamma ={10}^{-2}$, and ${\gamma }_{\mathrm{e}}=1/Q={10}^{-2}$. Color scale: white and red (light gray) regions correspond to $S_{\mathrm{sh}}\left(0\right)=0$ and $e^2\Gamma /4$, respectively.