1. Quantum pumping and rectification effects in interacting quantum dots Francesco Romeo In collaboration with : Dr Roberta Citro Prof. Maria Marinaro University of Salerno, Dipartimento di Fisica “E. R. Caianiello” Capri Spring School on Transport in Nanostructures 2009 - Capri, March 29 - April 5

2. Quantum pumping technique Electron pump in quantum dots (QD): the finite current problem at zero-phase Fisher-Lee & Brouwer-Wang approach Results for single level interacting and non- interacting QD Conclusions Outlines

3. Quantum pump: A device which generates a dc current by a periodic adiabatic variation of the system characteristics; Thouless (83) Quantization : Integral of current over a period is quantized in system with full bands (robust against disorder and interaction); Niu and Thouless (84) Mesoscopic systems : typically in quantum dots and semiconductors; Sharma & Brouwer 03 (Theory), Switkes (99), Watson (03) (Experiments) Quantum Pumping technique

4. Adiabatic Quantum Pumping P. Brouwer, PRB 1998 Current in weak pumping limit

5. An Adiabatic Quantum Electron Pump by Switkes et al. M. Switkes et al., Science 283, 1905 (1999) QD Problem: Finite current at zero phase

6. Possible theoretical explanations: Rectification of displacement currents Finite frequency effect (Wang) Non-equilibrium effect (+ interaction) F. Romeo, R. Citro and M. Marinaro, Phys. Rev. B 78, 245309 (2008). Finite zero-phase current

7. Hamiltonian of the system Hamiltonian model

8. Current and instantaneous scattering matrix Instantaneous pumped currents Time averaged pumped currents

9. Scattering matrix and Fisher-Lee relation (adiabatic theory) Fisher-Lee relation in adiabatic limit, i.e. Time derivative of the full Green’s function obtained from the Dyson equation by assuming the uncouple GF as time independent (no time modulation of the QD levels) D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981)

10. Pumping + rectification current Instantaneous pumped currents in terms of GF Linewidth and tunneling amplitudes

11. The wide band limit is assumed Pumping through a non-interacting single level QD Pumping term Rectification term Effective voltage Pumping term as in

12. A self-consistent treatment of the problem is required The renormalization of the dot level has been neglected in this approach The wide band limit is assumed Pumping through an “interacting” single level QD

13. Pumping cycle Results within the zero- temperature limit Total charge Pumping charge Rectified charge Energy scale Rectified charge in the strong interacting limit

14. Pumping cycle Results: time-linear variation of the tunneling phase Energy scale Total charge Rectified charge Pumping charge The pumping term (a) presents a sinusoidal behavior with respect to the pumping phase The rectification term (b) is proportional to the cosine of the pumping phase

15. Rectification contribution to the charge (weak pumping approximation) Analytical dependence of the rectification term vs pumping phase Ratchet-like term

16. We presented a Fisher-Lee based theoretical approach accounting for the finite zero-phase current detected in quantum pump devices. In this framework the rectification term is naturally included The proposed approach can be extended directly beyond the linear response regime (i.e. beyond the first order in the pumping frequency) The interaction effects can be included in the framework of a mean field treatment (slave boson etc) Energy current and noise can be analyzed within the mentioned theory Conclusions

17. Bibliography  F. Romeo, R. Citro, M. Marinaro; Phys. Rev. B 78, 245309 (2008)  R. Citro, F. Romeo; Phys. Rev. B 77, 193309 (2008)  F. Romeo, R. Citro, M. Marinaro; Phys. Rev. B 76, 081301(R) (2007)  R. Citro, F. Romeo; Phys. Rev. B 75, 073306 (2007)  R. Citro, F. Romeo, M. Marinaro; Phys. Rev. B 74, 115329 (2006)  R. Citro, F. Romeo; Phys. Rev. B 73, 233304 (2006)  F. Romeo, R. Citro; Eur. Phys. J. B 50, 483-489 (2006)  R. Citro, F. Romeo; Quantum pumping and rectification effects in interacting quantum dots, preprint (2009) ; arXiv:0903.2362v1

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19. We modulate only the interface barriers strength The energy levels on the QD are not modulated QD-based electron pump Fermi energy j-th QD energy level

20. Appendix (1): Delta term

21. Appendix (2): Pumping Formula Pumped current Energy current