Ja n 20 04 Shuttle instabilities : semiclassical phase analysis

We present a semiclassical analysis of the instability of an electron shuttle composed of three quantum dots: two are fixed and coupled via leads to electron resevoirs at µ L,R with µ L ≫ µ R , while the central dot is mounted on a classical harmonic oscillator. The semiclassical analysis, which is valid if the central dot oscillation amplitude is larger than the quantum mechanical zero point motion, can be used to gain additional insight about the relationship of resonances and instabilities of the device. In nano-electromechanical systems (NEMS) the electrical and mechanical properties are deeply interconnected. An archetypal NEMS device consists of a movable object connected to leads [1]. The charge distribution gives rise to an electrical force, which influences the mechanical dynamics, while the position of the movable object determines the tunneling rates from the leads and thus influences the electrical dynamics of the system. A current through the device can sustain mechanical oscillations even in the presence of damping. An interesting regime of transport arises when only one electron per cycle is transferred from the left to the right lead. Due to the position dependent tunneling amplitude the movable part gets charged when near to the left lead, then the electrostatic force pushes it towards the right lead where the now enhanced tunneling rate helps the release of the electron. We describe the electronic part with the density matrix formalism and couple the master equation to a classical equation of motion for the central dot position [1]. We perform a linear instability analysis of the system to pinpoint where the equilibrium solution for the system becomes unstable leading to Hopf bifurca-tions; the relative phase between charge, velocity and position of the unstable rotating solution shows the features of shuttling in agreement with the quantum-phase-space description [2]. The semiclassical approach is justified by the quantum-classical correspondence since the dot oscillations are bigger than the minimum quantum amplitude. However the mean field approach for the electric part neglects the effect of shot noise and the damping factor threshold is usually very much reduced compared to the quantum treatment. The semiclassical analysis is also much easier to handle numerically.