Discontinuous Euler instability in nanoelectromechanical systems

We investigate nanoelectromechanical systems near mechanical instabilities. We show that, quite generally, the interaction between the electronic and the vibronic degrees of freedom can be accounted for essentially exactly when the instability is continuous. We apply our general framework to the Euler buckling instability and find that the interaction between electronic and vibronic degrees of freedom qualitatively affects the mechanical instability, turning it into a discontinuous one in close analogy with tricritical points in the Landau theory of phase transitions.

Figure 1

(Color online) Sketch of a nanobeam (a) in the flat state and (b) the buckled state with two equivalent metastable positions of the rod (solid and dashed lines). An equivalent circuit of the embedded SET is shown in (c).

Figure 2

(Color online) (Meta)stable (solid blue lines) and unstable (dashed red lines) positions of the nanobeam vs scaled force $ϵ$ for (a) $\left|v_g\right|<v/2$, $v<1/2$, (b) $\left|v_g\right|<v/2$, $v>1/2$, (c) $v_g>v/2$, $v<1/2$ (for ${ϵ}_{+}>1$; a similar plot holds for ${ϵ}_{+}<1$), and (d) $v_g>v/2$, $v>1/2$, as indicated in the $v_g\text{−}v$ plane in (e). The dotted blue line is the result without electron-vibron coupling. Notation: ${ϵ}_{\pm }=2v_g\pm v$, $\widetilde{ϵ}=1/2+v_g/v$, $x_{+}^2=ϵ$, ${\tilde{x}}_{+}^2=ϵ-1$, and $x_{-}^2=\left(ϵ-\widetilde{ϵ}\right)/\left(1-1/2v\right)$. Grey indicates conducting regions.

Figure 3

(Color online) Conductance $G=RI/V$ in the $v_g\text{−}v$ plane for applied force (a) $ϵ\le 0$, (b) $ϵ=0.25$, (c) and (g) $ϵ=0.5$, (d) $ϵ=0.75$, (e) $ϵ=1$, and (f) and (h) $ϵ=1.25$, within (a)–(f) stability analysis and (g) and (h) full Langevin dynamics $\left(r={\gamma }_e=T=0.01\right)$. Color scale: $G=0\to 1/4$ from dark blue to white. Dotted lines delineate the Coulomb diamond for $g=0$.