Stochastic dynamics for a single vibrational mode in molecular junctions

We propose a very accurate computational scheme for the dynamics of a classical oscillator coupled to a molecular junction driven by a finite bias, including the finite-mass effect. We focus on two minimal models for the molecular junction: the Anderson-Holstein and two-site Su-Schrieffer-Heeger (SSH) models. As concerns the oscillator dynamics, we are able to recover a Langevin equation confirming what has been found by other authors with different approaches and indicating that quantum effects come from the electronic subsystem only. Solving numerically the stochastic equation, we study the position and velocity distribution probabilities of the oscillator and the electronic transport properties at arbitrary values of electron-oscillator interaction and gate and bias voltages. The range of validity of the adiabatic approximation is established in a systematic way by analyzing the behavior of the kinetic energy of the oscillator. Due to the dynamical fluctuations, at intermediate bias voltages, the velocity distributions deviate from a Gaussian shape and the average kinetic energy shows a nonmonotonic behavior. In this same regime of parameters, the dynamical effects favor conduction far from electronic resonances where small currents are observed in the infinite-mass approximation. These effects are enhanced in the two-site SSH model due to the presence of the intermolecular hopping t. For sufficiently large hopping with respect to tunneling on the molecule, small interaction strengths, and at intermediate bias (non-Gaussian regime), we point out a correspondence between the minima of the kinetic energy and the maxima of the dynamical conductance.