Influence of vibrational modes on the quantum transport through a nano-device

We use the recently proposed scattering states numerical renormalization group (SNRG) approach to calculate $I(V)$ and the differential conductance through a single molecular level coupled to a local molecular phonon. We also discuss the equilibrium physics of the model and demonstrate that the low-energy Hamiltonian is given by an effective interacting resonant level model. From the NRG level flow, we directly extract the effective charge transfer scale $\Gamma_{\rm eff}$ and the dynamically induced capacitive coupling $U_{\rm eff}$ between the molecular level and the lead electrons which turns out to be proportional to the polaronic energy shift $E_p$ for the regimes investigated here. The equilibrium spectral functions for the different parameter regimes are discussed. The additional phonon peaks at multiples of the phonon frequency $\w_0$ correspond to additional maxima in the differential conductance. Non-equilibrium effects, however, lead to significant deviations between a symmetric junction and a junction in the tunnel regime. The suppression of the current for asymmetric junctions with increasing electron-phonon coupling, the hallmark of the Franck-Condon blockade, is discussed with a simple framework of a combination of (i) polaronic level shifts and (ii) the effective charge transfer scale $\Gamma_{\rm eff}$.