Nonmonotonic Fano factor and backscattering effects in single-molecule junctions

Rate equations are a common tool to describe the transport properties of weakly coupled single-molecule junctions. Here, we study the physics of the Anderson-Holstein model at a single vibronic resonance. We derive conditions on the Franck-Condon factors that the resonance increases or decreases the stationary current thus causing negative differential conductance. The role of backscattering of charge at vibronic resonances is also investigated. In strongly asymmetrically coupled devices backscattering causes the resonance to split into two. In symmetrically coupled devices, the Fano factor shows nonmonotonicities, which are related to correlations of forward and backward scattering of charge.

Figure 1

(Color online) NDC in a Franck-Condon blockaded single-molecule junction with $\lambda =4$. (a) $dI/dV$ as a function of both applied voltages. The color code is black-gray-white for negative-zero-positive. (b) Magnified view of the first appearance of NDC in the stationary current for the resonance $|1;0⟩\mapsto |0;5⟩$ at the drain electrode.

Figure 2

(Color online) Backscattering effects due to strongly asymmetric coupling: (a) a full molecule with $\lambda =1$, where ${\Gamma }^{00}={\Gamma }^{10}$, ${\Gamma }^{11}=0$, and (b) a model system with cutoff at $q=1$. The dashed lines in graph (b) indicate $I\left(f=0\right)$, $\frac{3}{2}I\left(f=0\right)$, and $2I\left(f=0\right)$. The parameters are $\lambda =1$, $k_{\text{B}}T=0.03\hslash \omega$, and ${\Gamma }_{\text{L}}$ is varied from $100{\Gamma }_{\text{R}}$ (red, small peak) to the unphysical value of ${10}^8{\Gamma }_{\text{R}}$ (blue, large plateau), which, however, shows the effect discussed in the text.

Figure 3

(Color online) Fano factor for a symmetrically coupled device with ${\Gamma }^{00}=1={\Gamma }^{10}$ and ${\Gamma }^{11}$ varying (a) from 0 to 1 with $F\left(1\right)$ increasing and (b) from 1 to 3 with $F\left(1\right)$ decreasing—red to blue curves in each diagram. The solid lines are integer values of ${\Gamma }^{11}$. The voltage $V$ is measured with respect to the resonance being studied. In (c) and (d) the same situation is shown for ${\Gamma }^{10}=0.2{\Gamma }^{00}$. In (a), even low values of ${\Gamma }^{11}$ lead to an overshooting of $F$, which grows in magnitude for larger rates. For $\delta ={\Gamma }^{11}-{\Gamma }^{00}\to 0$, the curves become symmetric, and the overshooting peak is reduced in size. The maximum of the peak is not located at the resonance $eV=0$, except for ${\Gamma }^{11}={\Gamma }^{00}$, hinting at a more complex interplay of forward and backscattering dynamics.

Figure 4

(Color online) Fano factor for a single-molecule device and different electron-phonon coupling [$\lambda =0.5$ (red, upper curve), $\lambda =1$ (blue, middle curve), $\lambda =1.2$ (green, lower curve)—the temperature of the electrodes is $k_{\text{B}}T=0.025\hslash \omega$]. The overshooting at the first vibronic sideband is most pronounced for small $\lambda$ and at low bias, when only few vibronic states participate in the dynamics.