Simple models suffice for the single-dot quantum shuttle

A quantum shuttle is an archetypical nanoelectromechanical device, where the mechanical degree of freedom is quantized. Using a full-scale numerical solution of the generalized Master equation describing the shuttle, we have recently shown (Novotny et al 2004 Phys. Rev. Lett. 92 248302) that for certain limits of the shuttle parameters one can distinguish three distinct charge transport mechanisms: (i) an incoherent tunnelling regime, (ii) a shuttling regime, where the charge transport is synchronous with the mechanical motion, and (iii) a coexistence regime, where the device switches between the tunnelling and shuttling regimes. While a study of the crossover between these three regimes requires the full numerics, we show here that by identifying the appropriate timescales it is possible to derive vastly simpler equations for each of the three regimes. The simplified equations allow a clear physical interpretation, are easily solved and are in good agreement with the full numerics in their respective domains of validity.