Transport in Quantum Dots II

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Transport in Quantum Dots (II)

• 1) Review of Kondo effect in metals
• 2) Review of Coulomb blockade
• 3) Kondo effect in Qds
• 4) Kondo effect vs. Coulomb blockade
• 5) Coupled Qds

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Kondo effect in metals
Metal
Important A resonant state appears at EF
J. Kondo, Prog. Theor. Phys 32, 37 (1964)
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Kondo model -gt Anderson model

When Ni 1 the impurity Anderson model can be
mapped in the Kondo impurity model
(Schrieffer-Wolff transformation).
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Quantum Dot
Barriers (t')
Leads
Gate Potential (Vg)
Quantum Dot (U)
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Conductance
Vg
The conductance is defined as (0 bias)
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The effect of the temperature
Fermi distribution when T increase
Vg
Meir et al. Phys.Rev.B 23, 3048 (1991)
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Coulomb Blockade vs Kondo
Kondo regime (TltTk)
CB regime (TgtTk)
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Tk and experiments
The Tk for the Anderson impurity model is
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QD - Kondo effect
Coulomb Blockade valley
Kondo peaks
Decreasing Temp.
L.Kouwenhoven et al Phys.Today Jan.2001
D.Goldhaber-Gordon et al Nature 391, 156 (1998).
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Coupled Qds
Gate potential
Quantum dots
Barriers
Leads
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Dimer states
.t'
Vg
Vg
Vgt'
Vg
.t'
.t'
Vg-t'
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Exact diagonalization Dyson Eq.
1) Divide the system in two parts (cutting two
connections)
2) Solve the many-body cluster using exact
diagnalization technique (ground state
calculation)
3) re-establish the cutted connections using a
Dyson equation
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Some results
Density of states
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2QDs - Coulomb Blockade
Vg2
R
L
Vg1
Hubbard I calculation
F. Hofmann et al, Phys. Rev. B 51, 13872 (1995)
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2QDs - Kondo regime
Exact diagonalization calculations
H. Jeong, Science Vol 293, 2221 (2001)
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Conclusions

• 1) The discrete levels of the QDs can mimic the
  levels of the atoms. They are called artificial
  atoms.
• 2) The low energy physics (as Kondo effect) of
  atoms could be observed in Qds.
• 3) A collection of Qds can represent molecules

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..........
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Coulomb Diamond
Wider peaks in the conductance
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Transport in organic molecules
W. Liang et al, Nature Vol. 417, 725 (2002)
Leads
Small gap (1 nm) between the metal tips
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Kondo
No conductance
Coulomb blockade