A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča* The Kondo effect in multiple quantum dot systems and deformable molecules Department of Physics Faculty of Mathematics and Physics University of Ljubljana *

Outline (1) Conductance (2) Kondo in a single quantum dot (3) Methods (4) Double quantum dots (5) Triple quantum dots (6) Deformable molecules (7) Center-of-mass motion (8) Summary

Conductance

Conductance

Non-interacting systems : U=0

The Anderson model : U > 0

The Kondo effect in a quantum dot

1 “Ring system”

“Open system”

the GS energy of a large ring system is an universal function of flux the GS energy of a large ring system is an universal function of flux T. Rejec and A. Ramšak, PRB 68, (2003); (2003) IF open system is Fermi liquid IF open system is Fermi liquid

Linear conductance from the ground-state energy See also: J. Favand and F. Mila (Phys. J. 1998); O. Sushkov (PRB 2001); R. Molina et al. (PRB 2003)

Linear conductance from the ground-state energy

Aharonov – Bohm rings Broken time-reversal symmetry T. Rejec and A. Ramšak, PRB 68, (2003)

The Kondo effect in a quantum dot U=0 numerical tests…

The Kondo effect in a quantum dot: finite temperature U=0 high T low T

The Kondo effect in a quantum dot: finite temperature high T low T

The Kondo effect in a quantum dot: finite temperature high T low T

Fingerprints of Kondo…

Chan et al, Nanotechnology 15, 609 (2004) Vidan et al, Applied Phys. Lett. 85, 3602 (2004) Electrostatic gates QD Elzerman et al, PRB 67, (2003) QD Multiple quantum dot systems

Double quantum dot

J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 73, (R) (2006)

Double quantum dot 2 Kondo AFM 1 Kondo t t

Double quantum dot 2 Kondo AFM 1 Kondo t t

Double quantum dot 2 Kondo AFM 1 Kondo t t

Other topologies: local singlet vs the Kondo effect A.Ramšak, J. Mravlje, R. Žitko, and J. Bonča, quant-ph/

Thermal equilibrium: A-B spin corelations

Zero magnetic field, thermal equilibrium A B

Zero magnetic field, temperature A B

A B

A B

A B

Triple quantum dot

Deformable molecules e

e

H. Park et al. Nature 407 (2000) e Change in: local energy hopping matrix elements

…

Modeling

Isolated molecule: molecule attached to the leads: Lang & Firsov transformation: the result: Reduction of U and narrowing of the level-width A.C. Hewson and D.M. News J.Phys C 13 (1980) K. Schönhammer and O. Gunnarsson PRB 30 (1984) Old knowledge …

decrease negative U: A.Taraphder and P. Coleman, PRL 66, 2814 (1991).

J. Mravlje, A. Ramšak, and T. Rejec, PRB 72, (R) (2005); See also: P.S. Cornaglia, D.R. Grempel, and H. Ness, Phys. Rev. B 71, (2005), A. Mitra, I. Aleiner, and A.J. Millis, Phys. Rev. B 79, (2004).

Molecules with a center of mass motion J. Mravlje, A. Ramšak, and T. Rejec, submitted to PRB

Molecules with a center of mass motion

Friedel sum rule:

Molecules with a center of mass motion Friedel sum rule:

Molecules with a center of mass motion A B

Summary Linear conductance at T=0 can then be extracted from the GS energy: The Kondo effect: Low temperature destiny of quantum dots

ad summary…

Formulae are exact IF the system is Fermi liquid note: linear conductance zero temperature non-interacting single-channel leads

Fisher – Lee relation … Conductance formalisms non-equilibrium transport: T ≠ 0, V ≠ 0 U = 0 Landauer – Büttiker formula linear response regime: T ≠ 0, V ~ 0zero-temperature linear response: T = 0, V ~ 0 U ≠ 0 Meir – Wingreen formula In Fermi liquid systems Kubo formula

Proof of the method Step 1. Conductance of a Fermi liquid system at T=0 Kubo T=0 define (n.i.: Fisher-Lee) ‘Landauer’

Step 2. Quasiparticle hamiltonian (Landau Fermi liquid)

Step 3. Quasiparticles in a finite system N

Step 4. Validity of the conductance formulas