Vibrational sidebands and dissipative tunneling in molecular transistors

Transport through molecular devices with strong coupling to a single vibrational mode is considered in the case where the vibration is damped by coupling to the environment. We focus on the weak tunneling limit, for which a rate equation approach is valid. The role of the environment can be characterized by a frictional damping term $\mathcal{S}\left(\omega \right)$ and a corresponding frequency shift. We consider a molecule that is attached to a substrate, leading to a frequency-dependent frictional damping of the single oscillator mode of the molecule, and compare it to a reference model with frequency-independent damping featuring a constant quality factor Q. For large values of Q, the transport is governed by tunneling between displaced oscillator states, giving rise to the well-known series of the Frank-Condon steps, while at small Q, there is a crossover to the classical regime with an energy gap given by the classical displacement energy. Using realistic values for the elastic properties of the substrate and the size of the molecule, we calculate $I-V$ curves and find a qualitative agreement between our theory and recent experiments on ${\mathrm{C}}_{60}$ single-molecule devices.