Quantum pumping and rectification effects in

interacting quantum dotsFrancesco 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

Quantum pumping techniqueElectron pump in quantum dots (QD):

the finite current problem at zero-phaseFisher-Lee & Brouwer-Wang approachResults for single level interacting and

non-interacting QDConclusions

Outlines

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

Adiabatic Quantum Pumping

P. Brouwer, PRB 1998

Current in weak pumping limit

An Adiabatic Quantum Electron Pump by Switkes et al.

M. Switkes et al., Science 283, 1905 (1999)

QD

Problem: Finite current at zero phase

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

Hamiltonian of the system

Hamiltonian model

Current and instantaneous scattering matrix

Instantaneous pumped currents

Time averaged pumped currents

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)

Pumping + rectification current

Instantaneous pumped currents in terms of GF

Linewidth and tunneling amplitudes

The wide band limit is assumed

Pumping through a non-interacting single level QD

Pumping term

Rectification term Effective voltage

Pumping term as in

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

Pumping cycle

Results within the zero-temperature limit

Total charge

Pumping charge

Rectified charge

Energy scale

Rectified charge in the strong interacting limit

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

Rectification contribution to the charge (weak pumping approximation)

Analytical dependence of the rectification term vs pumping

phase

Ratchet-like term

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

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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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

Appendix (1): Delta term

Appendix (2): Pumping Formula

Pumped current

Energy current