Euler buckling instability and enhanced current blockade in suspended single-electron transistors

Single-electron transistors embedded in a suspended nanobeam or carbon nanotube may exhibit effects originating from the coupling of the electronic degrees of freedom to the mechanical oscillations of the suspended structure. Here, we investigate theoretically the consequences of a capacitive electromechanical interaction when the supporting beam is brought close to the Euler buckling instability by a lateral compressive strain. Our central result is that the low-bias current blockade, originating from the electromechanical coupling for the classical resonator, is strongly enhanced near the Euler instability. We predict that the bias voltage below which transport is blocked increases by orders of magnitude for typical parameters. This mechanism may make the otherwise elusive classical current blockade experimentally observable.

Figure 1

Sketch of the considered system: a suspended doubly clamped beam forming a quantum dot electrically connected to source and drain electrodes held at chemical potentials ${\mu }_{\mathrm{L}}$ and ${\mu }_{\mathrm{R}}$ by the bias voltage $V$, respectively. The beam is capacitively coupled to a metallic gate kept at a voltage $V_{\mathrm{g}}$, which induces a force $F_{\mathrm{e}}$ that attracts the beam toward the gate electrode. An additional externally controlled compressional force $F$ acts on the beam and induces a buckling instability.

Figure 2

Example of a solution $x(\delta )$ of the equation for dynamic equilibrium (21) for $n_0=0$ (dashed line), $n_0=1$ (solid line), and for $n_0=1$ and $\mathrm{\alpha ̃}=0$ (dotted line). In the figure, $\mathrm{\alpha ̃}={10}^{-3}$.

Figure 3

(a) Gap ${\Delta }_v$ and (b) gate voltage $v_{\mathrm{g}}$ (defined as the bias and gate voltages in reduced units at the apex of the Coulomb diamond, respectively) as a function of the scaled compression force $\delta =F/F_{\mathrm{c}}-1$. The red circles and blue squares are numerical results for $\mathrm{\alpha ̃}={10}^{-3}$ and ${10}^{-6}$, respectively, which are compared to the asymptotic behaviors (23) and (24) for forces below (solid line), above (dashed line), and in the vicinity (dotted line) of the critical force $F_{\mathrm{c}}$. (Inset) Gap ${\Delta }_v~1/\sqrt[3]{\mathrm{\alpha ̃}}$ from Eq. (23) as a function of $\mathrm{\alpha ̃}$ at the mechanical instability ($\delta =0$).

Figure 4

Mean-field current $I$ at zero temperature and for symmetric coupling to the leads (${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}=\Gamma /2$) as a function of bias $v$ and gate voltage $v_{\mathrm{g}}$ (measured in units of the elastic energy $E_{\mathrm{E}}^0$). The (scaled) compression force $\delta$ increases from (a) to (g). Notice that the scale of the $v_{\mathrm{g}}$ axis is different in (e), (f), and (g) and in (a)–(d). The red dashed lines indicate the position of the Coulomb diamond in the absence of electromechanical coupling ($F_{\mathrm{e}}=0$). In the figure, $\mathrm{\alpha ̃}={10}^{-6}$, and dark blue and white regions correspond to $I=0$ and $I=e\Gamma /4$, respectively.

Figure 5

Current $I$ at the apex of the Coulomb diamond as a function of bias $v$ scaled by the energy gap ${\Delta }_v$ for various values of $\mathrm{T̃}/{\Delta }_v$ and for compression forces in the vicinity of the buckling instability ($\left|\delta \right|⩽\sqrt[3]{\mathrm{\alpha ̃}}$). In the figure, only the temperature-induced fluctuations are considered. (Inset) Same as the main figure for compression forces far from the buckling instability ($\left|\delta \right|\gg \sqrt[3]{\mathrm{\alpha ̃}}$).

Figure 6

Current $I$ at the apex of the Coulomb diamond for $\delta =0$ as a function of $v/{\Delta }_v$ for $\mathrm{\alpha ̃}={10}^{-6}$ and $\mathrm{T̃}/{\Delta }_v=0.1$. (Solid lines) Nonequilibrium Langevin dynamics for $\frac{{\gamma }_{\mathrm{e}}}{{\omega }_0/\Gamma }=1$, ${10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, and 0, from the highest to the lowest curve at large bias. (Dashed-dotted line) Fully adiabatic limit ($\frac{{\gamma }_{\mathrm{e}}}{{\omega }_0/\Gamma }=\infty$), i.e., current only including thermal fluctuations (cf. dashed-dotted curve in Fig. 5). In our numerical calculations, we used ${\omega }_0/\Gamma ={10}^{-2}$.

Figure 7

(a) Current $\mathcal{I}$ as a function of $x$ [Eq. (13)] and (b) effective potential $v_{\mathrm{eff}}(x)$ [Eq. (25)] for bias voltages below (dashed line), above (dotted line), and at (solid line) the energy gap ${\Delta }_v$. The parameters are the same as in Fig. 6, i.e., $\delta =0$, $\mathrm{\alpha ̃}={10}^{-6}$, $\mathrm{T̃}/{\Delta }_v=0.1$, and $v_{\mathrm{g}}=-3/4\sqrt[3]{\mathrm{\alpha ̃}}$.

Figure 8

(a) Gap ${\Delta }_v$ and (b) gate voltage $v_{\mathrm{g}}$ at the apex of the Coulomb diamond for $\mathrm{\alpha ̃}={10}^{-6}$ as a function of the scaled compression force $\delta$ for increasing values of $f_N$ [from top to bottom and bottom to top at $\delta =0$ in (a) and (b), respectively]. The blue squares ($f_N=0$) correspond to the numerical results of Fig. 3, while the solid lines result from the harmonic approximation (see text). (Inset) Gap ${\Delta }_v~1/f_N^{2/3}$ from Eq. (29) as a function of $f_N$ at the mechanical instability ($\delta =0$).

Figure 9

Coordinate system used to describe the elastic properties of the rod.

Figure 10

Sketch of the Coulomb diamond for a SET linearly coupled to a harmonic oscillator, delimited by the solid lines at $v_{+}=2{\mathrm{ṽ}}_{\mathrm{g}}+3/2v^{\prime }(\mathrm{x̄})$ and $v_{-}=-2{\mathrm{ṽ}}_{\mathrm{g}}-1/2v^{\prime }(\mathrm{x̄})$. The gray area indicates the region where current can flow. The dashed lines indicate the location of the Coulomb diamond without electromechanical coupling ($F_{\mathrm{e}}=0$), delimited by $v=\pm 2{\mathrm{ṽ}}_{\mathrm{g}}$, which defines the three regions of the $v$-${\mathrm{ṽ}}_{\mathrm{g}}$ plane discussed in the text.