Franck-Condon Blockade and Giant Fano Factors in Transport through Single Molecules

We show that Franck-Condon physics leads to a significant current suppression at low bias voltages (termed Franck-Condon blockade) in transport through single molecules with strong coupling between electronic and vibrational degrees of freedom. Transport in this regime is characterized by remarkably large Fano factors (${10}^2–{10}^3$ for realistic parameters), which arise due to avalanchelike transport of electrons. Avalanches occur in a self-similar manner over a wide range of time scales, leading to power-law dependences of the current noise on frequency and vibrational relaxation rate.

Figure 1

(a) Current-voltage characteristics for intermediate ($\lambda =1$) and strong ($\lambda =4$) electron-phonon coupling for $\epsilon =0$ and $k_{B}T=0.05\hslash {\omega }_0$ with (eq.) and without (uneq.) relaxation of phonons. We find that strong electron-phonon coupling leads to a significant current suppression at low bias voltages. This Franck-Condon blockade arises from the behavior of the Franck-Condon rates for phonon transitions $q_1\to q_2$ plotted in (b). The rates ${\Gamma }_{q_1{q}_2}$, shown for $\lambda =1$ (left) and $\lambda =4$ (right), are given in units of $2\pi \rho t_0^2/\hslash$. For strong electron-phonon coupling, transitions between low-lying phonon states are exponentially suppressed. The corresponding current suppression cannot be lifted by gate voltage, which may serve as a fingerprint of FC blockade. This is depicted in the plot of $dI/dV$ in the $V\mathrm{\text{−}}{V}_g$ plane for unequilibrated phonons with $\lambda =4$ (c). The case of intermediate coupling with $\lambda =1$ (d) is shown for comparison.

Figure 2

Zero-frequency (excess) Fano factor as a function of bias voltage with $\lambda =4$ for $\epsilon =0$ and $k_{B}T=0.05\hslash {\omega }_0$. Results are shown for equilibrated phonons ($\tau =0$) and unequilibrated phonons ($\tau =\infty$). We find that the combined effect of strong electron-phonon coupling and weak phonon relaxation leads to giant Fano factors in the Franck-Condon blockade regime.

Figure 3

Number of transmitted electrons $N$ and number of excited phonons $q$ as functions of time, obtained by MC simulation for $T=0.05\hslash {\omega }_0$, $V=3\hslash {\omega }_0/e$, $\epsilon =0$, $\tau =\infty$, and $\lambda =4$. The upper plot shows a magnification of the marked region in the lower plot. The results show long periods of time without current flow when the system assumes the phononic ground state, and avalanchelike current flow accompanied by phonon excitations on much shorter time scales. Self-similarity of the curves is found when magnifying the avalanche phases.

Figure 4

Noise power in the Franck-Condon blockade regime as a function of (a) frequency; (b) phonon relaxation rate for $\epsilon =0$, $\lambda =4$, and $k_{B}T=0.05\hslash {\omega }_0$. In (a) the total current noise is plotted for unequilibrated phonons. In (b) the zero-frequency noise is plotted for the bias voltage $eV=3\hslash {\omega }_0$. We find (approximate) power-law behavior $S(\omega )\sim {\omega }^{-\alpha }$ and $S(0)\sim {\tau }^{1/\alpha }$ with $\alpha \approx 1/2$, arising from the self-similar avalanches over a wide range of time scales.