A. Ram

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The Kondo effect in multiple quantum dot systems
and deformable molecules
http//gawain.elte.hu/mcrtn/
A. Ramšak J. Mravlje T. Rejec R. Žitko J.
Bonca

Department of Physics Faculty of Mathematics and
Physics University of Ljubljana
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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

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Conductance
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Conductance
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Non-interacting systems U0
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Non-interacting systems U0
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The Anderson model U gt 0
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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The Kondo effect in a quantum dot
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Ring system
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Open system
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Open system
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IF open system is Fermi liquid
the GS energy of a large ring system is an
universal function of flux
T. Rejec and A. Ramšak, PRB 68, 033306 (2003) 68
035342 (2003)
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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)
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Linear conductance from the ground-state energy
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Linear conductance from the ground-state energy
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Aharonov Bohm rings
Broken time-reversal symmetry
T. Rejec and A. Ramšak, PRB 68, 033306 (2003)
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The Kondo effect in a quantum dot
numerical tests
U0
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The Kondo effect in a quantum dot finite
temperature
low T
U0
high T
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The Kondo effect in a quantum dot finite
temperature
low T
high T
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The Kondo effect in a quantum dot finite
temperature
low T
high T
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Fingerprints of Kondo
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Fingerprints of Kondo
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Fingerprints of Kondo
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Fingerprints of Kondo
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Multiple quantum dot systems
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Double quantum dot
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Double quantum dot
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Double quantum dot
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J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B
73, 241305(R) (2006)
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Double quantum dot
1 Kondo
AFM
2 Kondo
t
t
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Double quantum dot
1 Kondo
AFM
2 Kondo
t
t
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Double quantum dot
1 Kondo
AFM
2 Kondo
t
t
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Other topologies local singlet vs the Kondo
effect

1. Ramšak, J. Mravlje, R. Žitko, and J. Bonca,
   quant-ph/0608065.

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Thermal equilibrium A-B spin corelations
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Zero magnetic field, thermal equilibrium
A
B
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Zero magnetic field, temperature
A
B
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Zero magnetic field, temperature
A
B
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Zero magnetic field, temperature
A
B
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Zero magnetic field, temperature
A
B
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Triple quantum dot
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Deformable molecules
e
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Deformable molecules
e
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Deformable molecules
e

• Change in
• local energy
• hopping matrix elements

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Modeling
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Old knowledge
Isolated molecule
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)
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decrease

• negative U
• Taraphder and P. Coleman,
• PRL 66, 2814 (1991).

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J. Mravlje, A. Ramšak, and T. Rejec, PRB 72,
121403(R) (2005) See also P.S. Cornaglia, D.R.
Grempel, and H. Ness, Phys. Rev. B 71, 075320
(2005), A. Mitra, I. Aleiner, and
A.J. Millis, Phys. Rev. B 79, 245302 (2004).
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Molecules with a center of mass motion
J. Mravlje, A. Ramšak, and T. Rejec, submitted to
PRB
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Molecules with a center of mass motion
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Molecules with a center of mass motion
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Molecules with a center of mass motion
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Molecules with a center of mass motion
Friedel sum rule
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Molecules with a center of mass motion
Friedel sum rule
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Molecules with a center of mass motion
A
B
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Molecules with a center of mass motion
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Molecules with a center of mass motion
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Summary

• Linear conductance at T0 can then be extracted
  from the GS energy

• The Kondo effect Low temperature destiny of
  quantum dots

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ad summary
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Formulae are exact IF the system is Fermi liquid

• note
• linear conductance
• zero temperature
• non-interacting single-channel leads

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Conductance formalisms
U 0
non-equilibrium transport T ? 0, V ? 0
Landauer Büttiker formula
In Fermi liquid systems
Fisher Lee relation
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Proof of the method
Step 1. Conductance of a Fermi liquid system at
T0
Kubo
T0
define (n.i. Fisher-Lee)
Landauer
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Step 2. Quasiparticle hamiltonian (Landau Fermi
liquid)
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Step 3. Quasiparticles in a finite system
N
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Step 4. Validity of the conductance formulas
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