Incoherent dynamics of vibrating single-molecule transistors

We study the tunneling conductance of nanoscale quantum ``shuttles'' in connection with a recent experiment [H. Park et al., Nature $407,$ 57 (2000)] in which a vibrating ${\mathrm{C}}_{60}$ molecule was apparently functioning as the island of a single electron transistor (SET). While our calculation starts from the same model of previous work [D. Boese and H. Schoeller, Europhys. Lett. $54,$ 668 (2001)] we obtain quantitatively different dynamics. Calculated $I\ensuremath{-}V$ curves exhibit most features present in experimental data with a physically reasonable parameter set, and point to a strong dependence of the oscillator's potential on the electrostatics of the island region. We propose that in a regime where the electric field due to the bias voltage itself affects island position, a ``catastrophic'' negative differential conductance (NDC) may be realized. This effect is directly attributable to the magnitude of overlap of final and initial quantum oscillator states, and as such represents experimental control over quantum transitions of the oscillator via the macroscopically controllable bias voltage.