Kondo effect

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Lecture 17 Kondo effect
Jun Kondo's paper "Resistance Minimum in Dilute
Magnetic Alloys" was published in Progress of
Theoretical Physics 32 (1964) p.37. Although
forty years have passed since then, the
importance of this work has not diminished, but
continues to increase. Kondo solved the
long-standing mystery of resistance minimum
phenomenon in his study, thereby opening a door
to fundamental and universal physics this is now
known as the Kondo effect. 
Todays lecture Kondo effect in SET transistors
and in nanowires
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Kondo effect in bulk metallic samples
Fe or Co impurity, which is magnetic, i.e. it has
a localized spin
a cloud of conduction electrons at the Fermi level
Au
?K
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Exchange interaction between spins
We assume a Hamiltonian for two identical
electrons of the form
which does not depend on the spin. The
Hamiltonian is symmetric with respect to the
particle indices 1 and 2. The solutions of the
stationary Schrödinger equation
for the orbital parts of the wave function can be
classified into symmetric and anti-symmetric with
respect to swapping     and     this is
because we have
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Exchange interaction between spins
which means that the permutation operator     
commutes with the Hamiltonian. The eigenstates of
    can therefore be chosen such they are also
simultaneous eigenstates of      which are
symmetric and antisymmetric wave functions with
respect to swapping     and    .
Since the total wave function (orbital times
spin) must be antisymmetric, this means that for
energy levels corresponding to symmetric orbital
wave functions lead to spin singlets with total
spin        . Energy levels corresponding to
anti-symmetric orbital wave functions lead to
spin triplets with total spin        . Even
though there is no spin-dependent interaction
term in the Hamiltonian, the spin and the
possible energy values are not independent of
each other!
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Exchange interaction between spins
Assume we treat the interaction term
                in the Hamiltonian Eq.  as a
perturbation.
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Kondo effect in bulk metallic samples
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Kondo resonance occurs at the Fermi energy
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Kondo resonance observed with STM
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Quantum dot devices
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Kondo effect in quantum dots
N-odd
N-even
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Kondo effect in quantum dots Kouwenhoven et al,
1998
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Kondo effect in nanowires absence of size
dependence
page 2053
w38 nm or lager
I
I-
Au Fe
V
V-
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Kondo effect in nanowires absence of size
dependence
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• In mesoscopic samples (nanowires and thin films)
• the resistance increases with decreasing
  temperature
• due to the following three reasons
• Weak localization
• Electron-electron interaction
• Kondo effect
• Other effects can also be present (but not
  important in our present discussion)
• a. Anderson or strong localization
• b. Coulomb gap
• c. Tomonaga-Luttinger phenomena
• d. Hopping effect (activation transport)

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Nanowires contributions from weak localization
and e-e interactions
pure electron-electron interaction effect is seen
in Fig.2a, where the weak localization is
suppressed by the magnetic field and the effect
of magnetic impurities is subtracted using data
from a very wide sample
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Subtracting the electron-electron interaction
contribution
The thermal diffusion length, LT, determines the
dimensionality of the sample. It is considered 1D
if the width WltLT.
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Magnetic field effect confirms Kondo physics