Semiclassical dynamics of nanoelectromechanical systems

We investigate the dynamics of a single phonon (oscillator) mode linearly coupled to an electronic few-level system in contact with external particle reservoirs (leads). A stationary electronic current through the system generates nontrivial dynamical behavior of the oscillator. Using Feynman-Vernon influence functional theory, we derive a Langevin equation for the oscillator trajectory that is nonperturbative in the system-leads coupling and from which we extract effective oscillator potentials and friction coefficients. For the two simplest cases of a single and two coupled electronic levels, we discuss various regimes of the oscillator dynamics.

Figure 1

Sketch of the AHM model. Two leads (gray bars) with chemical potentials ${\mu }_{\text{L}}$, ${\mu }_{\text{R}}={\mu }_{\text{L}}-e{V}_{\text{Bias}}$ embed the dot level which couples to an oscillator. Here $V_{\text{Bias}}$ is the bias voltage and $e$ the electron charge. The level energy ${\epsilon }_{\text{d}}$ is shifted by the oscillator position $x$, the shifted level energy ${\epsilon }_x={\epsilon }_{\text{d}}-\lambda x$ is stationary, if the oscillator force $F_{\text{Hooke}}=m{\omega }_0^{2}x$ and the electron force $F_{\text{e}}=\lambda ⟨\widehat{n}\left[x\right]⟩$ are in balance. Here $m$ and ${\omega }_0$ describe the oscillator mass and angular frequency, and $\lambda \left(g\right)$ denotes the coupling constant, cf. Eq. (10). For sufficiently small tunneling rates ${\Gamma }_{\text{L}}$, ${\Gamma }_{\text{R}}$ the stable stationary level energies (dashed lines) are located at $x=0$, $g{l}_0{\Gamma }_{\text{L}}/\Gamma$, and $g{l}_0$. The unstable points of ${\epsilon }_x$ (dotted lines) are located around the chemical potentials.

Figure 2

(Color online) (Left) Density plot of the effective oscillator potential $U_{\text{eff}}$ as a function of oscillator position $⟨x⟩$ and $V_{\text{Bias}}$ in units of ${\omega }_0$ and $l_0$ with the parameters ${\Gamma }_{\text{L},\text{R}}/{\omega }_0=0.7$, ${\epsilon }_{\text{d}}/{\omega }_0=2.9$, $\beta {\omega }_0=10$, and $g=2.4$. For increasing bias voltage, the effective potential shows two, three, two, and one minima. This different regions are separated by white lines at $V_{\text{Bias}}/{\omega }_0=2.04/2.58/2.64$. (Right) $U_{\text{eff}}$ and the two contributions $F_{\text{Hooke}}$, $F_{\text{e}}$ to the force $F_{\text{eff}}$ at bias voltages (symmetric choice) $V_{\text{Bias}}/{\omega }_0=0.0/2.3/5.0$. $U_{\text{eff}}$ exhibits extrema where the two force contributions (gray and dashed line) are in balance. The width of the center plateau in the electronic force contribution $F_{\text{e}}\propto ⟨n⟩$ grows with increasing bias voltage $V_{\text{Bias}}$, cf. Fig. 1 for the intermediate occupation $⟨n⟩$. For sufficiently high bias voltage, only one minimum remains.

Figure 3

(a) Effective oscillator potentials, (b) frictions $A\left(x\right)$, and the phase space portraits without (c) and with (d) friction in units of ${\omega }_0$ and $l_0$ with the parameters ${\Gamma }_{\text{L},\text{R}}/{\omega }_0=0.7$, ${\epsilon }_{\text{d}}/{\omega }_0=3.0$, and $g=2.45$ at zero temperature. From the bottom to the top the values of the bias voltage (symmetric choice) read $V_{\text{Bias}}/{\omega }_0=0.0/2.2/6.0$ in correspondence to the three regions in Fig. 2 (right). The peaks of the friction are located at $x=\left[{\epsilon }_{\text{d}}\mp V_{\text{Bias}}/2\right]/\lambda$ where the shifted level energies ${\epsilon }_x$ are in resonance with the chemical potentials ${\mu }_{\text{L},\text{R}}$, cf. Fig. 1.

Figure 4

Illustration of three processes that transfer energy from or to the oscillator. Graph A depicts the emission of energy by the electronic system, resulting in negative damping of the oscillator. In contrary, graph B and C show processes of energy absorption, causing positive damping. $\Delta$ denotes the energy that electrons must expend to enter an empty state in the right lead.

Figure 5

Examination of the DQD system for various tunnel couplings $T_c$, increasing from left to right. Row A shows the results of a fixed-point analysis. $\Delta$ corresponds to the determinant and $\tau$ to the trace of the Jacobian matrix. Rows B and C display phase space portraits without and with friction. Explicit parameters are ${\left|T_c\right|}^2=0.4;0.43;0.49;1.0{\omega }_0^2$. We used equal rates ${\Gamma }_{\text{L}}={\Gamma }_{\text{R}}=1.5{\omega }_0$. For the chemical potentials, we assumed ${\mu }_{\text{L}}=1\hslash {\omega }_0$ and ${\mu }_{\text{R}}=-5\hslash {\omega }_0$. The dimensionless coupling constant is chosen as $g=2.5$, and the internal bias voltage as $V_{\text{int}}=5\hslash {\omega }_0$ whereas ${\nu }_{\text{L}}=-{\nu }_{\text{R}}=e{V}_{\text{int}}/2$.