Charge transport in a quantum electromechanical system

We describe a quantum electromechanical system comprising a single quantum dot harmonically bound between two electrodes and facilitating a tunneling current between them. An example of such a system is a fullerene molecule between two metal electrodes [Park et al., Nature 407, 57 (2000)]. The description is based on a quantum master equation for the density operator of the electronic and vibrational degrees of freedom and thus incorporates the dynamics of both diagonal (population) and off diagonal (coherence) terms. We derive coupled equations of motion for the electron occupation number of the dot and the vibrational degrees of freedom, including damping of the vibration and thermo-mechanical noise. This dynamical description is related to observable features of the system including the stationary current as a function of bias voltage.

Figure 1

Schematic representation of tunneling between a source and a drain through a quantum dot. The dot is harmonically bound and vibrational motion can be excited as electrons tunnel through the system.

Figure 2

Average number of electron, phonon, and current through the dot against bias voltage with $\hslash {\omega }_0=5\mathrm{meV}$, $\hslash {\omega }_I-\eta =15\mathrm{meV}$, $k_{B}T=0.13\mathrm{meV}$, and $\hslash {\gamma }_L=\hslash {\gamma }_R=2\mu \mathrm{eV}$ for $\lambda =0.3$: (a), (b), (c) without damping and (d), (e), (f) with damping $\kappa ={0.3}_{\gamma L}$. We assume ${\mu }_L=-{\mu }_R=e{V}_{\mathrm{bias}}∕2$.

Figure 3

Steady state current for different values of $\lambda$ with damping $\kappa =0.3{\gamma }_L$; the electronic and phonon temperatures are both $1.5\mathrm{K}$.

Figure 4

Steady state current for different values of $\lambda$ and damping $\kappa =0.3{\gamma }_L$: the electronic temperature is $1.5\mathrm{K}$, and the phonon temperature is $116\mathrm{K}$, which is chosen to be $2\hslash {\omega }_0∕k_B$ in order to make manifest the step at a smaller bias voltage.

Figure 5

Difference in the correlation function for different values of $\lambda$ with damping $\kappa =0.3{\gamma }_L$.

Figure 6

Difference in the correlation function for different values of $\kappa$ with damping $\kappa =0.3{\gamma }_L$. The plot starts at a value $\kappa \ne 0$.