1. Quantum Theory of Shuttling Instability Tomáš Novotný §¶, Andrea Donarini §, and Antti-Pekka Jauho § § Mikroelektronik Centret, Technical University of Denmark Kgs. Lyngby, Denmark ¶ Department of Electronic Structures, Charles University Prague, Czech Republic See also: Phys. Rev. Lett. 90 (25), 256801 (2003); cond-mat/0301441

2. 2 Shuttling Instability in a nano-electromechanical system (NEMS) L. Y. Gorelik et al., Phys. Rev. Lett. 80, 4526 (1998)

3. 3 Important aspects: electric field acting on the charged grain, exponential position dependence of the tunnelling amplitude (“feedback”). These lead to the development of the nonlinear limit cycle dynamics for small enough friction – called shuttling. Theoretical description - coupled mechanical equation of motion for the position coordinate and master equation for the extra electron occupation probabilities ( ): are given by the standard expression for the Coulomb blockade (Fermi Golden Rule).

4. 4 Quantization of the current in terms of mechanical frequency which can be much higher than the bare tunneling rate Current vs. voltage and inverse friction: Coulomb staircase even for a symmetric junction above threshold voltage (or below threshold friction) ( )

5. 5 Further development of the quasiclassical theory inclusion of the shot noise (possible effect of hysteresis) – A. Isacsson et al., Physica B 255, 150 (1998), large damping – T. Nord et al., Phys. Rev. B 65, 165312 (2002), gate effects – N. Nishiguchi, Phys. Rev. B 65, 35403 (2001), coherent electronic transfer for a small movable quantum dot (further miniaturization leads to a discrete spectrum of the grain) – D. Fedorets et al., Europhys. Lett. 58, 99 (2002). Last two papers studied the single molecular transistor experiment by H. Park et al., Nature 407, 57 (2000) – a plausible IV-curve predicted by shuttling assumption But there is a competing theory of incoherent phonon-assisted tunnelling by D. Boese and H. Schoeller, Europhys. Lett. 54, 668 (2001).

6. 6 Quantum Theory of Shuttling When the size of a movable quantum dot is further decreased so that its zero-point amplitude becomes comparable with the electron tunnelling length the quantum mechanical treatment is necessary. The very first attempt was by A. D. Armour and A. MacKinnon, Phys. Rev. B 66, 35333 (2002). A modified model of an array of three quantum dots with the central one movable – technical simplification but physical complication (two non-trivial effects mixed: shuttling and interdot coherence). Description by generalized master equation (GME) for very high bias due to S. A. Gurvitz et al., Phys. Rev. B 53, 15932 (1996). Shuttling behaviour inferred indirectly from the current curve and present only for a special alignment of dots’ energies.

7. 7

8. 8 Theory for the original setup, i.e. a single dot and leads are Hamiltonians of a linear harmonic oscillator, generic heat bath and generic Ohmic coupling between the oscillator and the bath, respectively. We again work in the large bias limit. Electric field E and bias V are not related (should be by a selfconsistent electrostatics calculation).

9. 9, Then we can obtain the following GME for the reduced density matrix of the “System” one electronic level (diagonal elements) + oscillator degrees of freedom:

10. 10 The stationary current reads (due to the charge conservation) We use a translationally invariant form of the damping kernel which does not in general conserve positivity. The breaking of positivity is however irrelevant and occurs only for the high friction regime (with no shuttling). We also evaluate the charge resolved Wigner phase space distribution functions (quasiprobabilities) – they provide us with information about the charge and the oscillator at the same time and enable us to observe the shuttling transition directly.

11. 11 Note on the numerics (Prof. T. Eirola, Helsinki University of Technology) We solve the stationary equations truncated at large N by an iterative method known as the preconditioned Arnoldi scheme (modification of the Lanczos algorithm for non-hermitian matrices). The preconditioning is crucial for the convergence of the method. We use the inversion of the Sylvester part of the problem as the preconditioner. It is very efficient compared to direct methods – low memory requirements and fast enough. We achieve a solution for up to N=100 oscillator states on a normal PC within minutes. In a direct method we would have a supermatrix of the dimension 2*N*N = 20 000 (huge memory requirements and slow).

12. 12 Current-damping curve for zero temperature and

13. 13 Wigner functions for

14. 14 Wigner functions for

15. 15 Wigner functions for

16. 16 Wigner functions for

17. 17 Results We observe the quantum analogy of the shuttling transition as a function of the mechanical damping (our control parameter). High damping corresponds to the tunnelling regime with the current proportional to the bare tunnelling rates. The oscillator is in the thermal state (ground state at zero temperature) and the fluctuations only renormalize the bare rates. Low damping corresponds to the shuttling regime with the current independent of the tunnelling rates but instead proportional to the frequency The oscillator orbits almost classically, carries the charge from the left to the right lead and returns empty. Here, the transition is smeared into a crossover due to the presence of various sources of noise, namely the electrical shot noise (due to discrete change of the charge state) and the mechanical zero-point and thermal noise (Langevin force due to the bath).

18. 18 Analysis of the purely quantum shuttling for E = 0 For zero electric field (still very high bias in the leads) there is quasiclassically no shuttling. However, we do observe shuttling-like behaviour for E = 0 in the IV-curve as well as in the phase space picture for small enough damping. The shuttling is less pronounced, because it is driven exclusively by the noise, in particular the “quantum component of the shot noise”. The effect is generated by asymmetric quantum heating of the oscillator due to electronic transfer.

19. 19 If we rewrite the GME directly for the Wigner functions (for zero mechanical damping and in the second order in the Planck constant) we get: The red terms yield charge-resolved diffusion constants proportional to the square of the Planck constant (classically proportional to the temperature – therefore the name quantum heating).

20. 20 Summary We studied the shuttling instability in the quantum regime ( ). The transition as a function of mechanical damping exists even in the quantum regime in analogy to the classical case The transition is not sharp but is smeared into crossover by noise The noise is important – the position of the crossover is changed substantially from the classical value (4-5 times higher damping) Quantum noise can drive the transition even for zero electric field (purely quantum effect) Increasing temperature faciliates the transition due to stronger driving noise (however only a part of the story)

21. 21 Problems & Outlook Two main problems: 1.How to get beyond Markovian approximation? Infinite bias assumption ensures Markovian dynamics; still some problems even within the Markovian description (positivity of the damping kernel) For a finite bias we have to use non-Markovian description (in any dialect). However, the systems dynamically switches between weak and strong coupling in electronic transfer while shuttling – solution?! 2.More complex quantities than just mean value of current, e.g. current noise (to distinguish between shuttling and tunnelling) How to evalute quantities containing not only system operators?