Non equilibrium noise as a probe of the Kondo effect in mesoscopic wires

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Non equilibrium noise as a probe of the Kondo
effect in mesoscopic wires

• Eran Lebanon
• Rutgers University
• with Piers Coleman
• arXiv cond-mat/0501001

DOE grant DE-FE02-00ER45790
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Outline

• Motivation relevance of magnetic impurities in
  diffusive wires
• Enhanced inelastic scattering at and
  noise peak
• PT in impurity concentration
  equivalence to a quantum dot
• Calculation schemes -RPT
• -NCA, coupling scaling.
• Break down of the perturbation theory.

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SC
Vs-w
Diffusive metallic wire
V
Anomalous collision integral Kernel!
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• Consistent with later experimental observation
• Energy relaxation is stronger for the less pure
  metals
• The energy relaxation is quenched by a magnetic
  field

Quantitative comparisons to experimental
data Kroha Zawadowski PRL 88, 176803
(2002) Göppert Grabert PRB 64, 033301 (2001)
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What happens when the bias is reduced to the
Kondo temperature and below,VTK and
VltltTkwhere Kondo correlations become strong?
Experimenttheory In gold and copper wires the
energy relaxation is dominated by magnetic
impurity mediated interaction and not by direct
electron-electron interaction even for dilute
doping (1ppm)
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Thouless Energy
Energy relaxation Rate
Macroscopic wire Local Equilibrium
Mesoscopic wire Elastic distribution
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0ltxlt1 distance from the left electrode divided by
length of wire L
Mesoscopic wire Elastic distribution function
Macroscopic wire Local Equilibrium distribution
Nagaev, 92
Noise probes inelastic processes.
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How does ?in-1 due to magnetic impurity mediated
interaction depends on bias?
Small Bias VltltTK
At T0, V0
The impuritys
magnetic moment
is screened completely and the Kondo
singlet is fully developed. The physics is
described by a local Fermi liquid fixed point
there are no inelastic scatterings at low
energies. For small V The energy relaxation
increases with the bias and is proportional to V2
.
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How does ?in-1 due to magnetic impurity mediated
interaction depends on bias?
Large Bias VgtgtTK
The infra red singularities of perturbation
theory are cutoff by the bias. The problem
becomes a weak coupling problem with an effective
coupling Jeff ln-1 ( eV / kBTK ). The
relaxation rate decays with V
like a
polynomial of Jeff ln-1 ( eV / kBTK ).
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?in-1 ? ?
VltltTK VgtgtTK
Enhancement of the inelastic scattering rate for
intermediate biases VTK.
Non equilibrium reminiscent of the equilibrium
dephasing rate peak
This will be manifested in a peak in both the
noise curve and the intensity of the distribution
smearing as functions of the bias.
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Model
Operators creates an electron on i-th
impurity, occupation of i-th impurity,
conduction electron field
operator. Parameters impurity
orbital energy, on-site repulsion,
mixing amplitude. LMR
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Model
creates electron on i-th impurity,
occupation of i-th impurity,
conduction electron field operator.
impurity orbital energy,
on-site repulsion, mixing amplitude. LMR
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Boltzmann equation
Baym Kadanoff

• Fourier transform with respect to relative
  coordinates
• Gradient expansion keeping
• Summation over the momentum
• The current in the system is diffusive

For dilute magnetic impurities
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Perturbation theory in the impurity concentration
without impurities
The cores. noise
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Equivalence to a quantum dot problem
For ninltlt1 perturbation theory in ci.
t?fx(0) equivalent to quantum dot t-matrices.
Coupled by and
to electrodes at chemical potentials
µL and µR.

• Schematically
• ninltlt1 electrons do not scatter inelastically
  twice
• Diffusion communicates the distribution from the
  leads

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Schemes for Calculation of t?fx(0)

• Non equilibrium Kondo problem - an open problem.
  No reliable approach to describes the crossover
  regime VTK.
• Analytically for VltltTK extension of Hewsons
  renormalized perturbation theory (RPT) to weak
  non-equilibrium ? Perturbation theory in e/TK,
  T/TK and V/Tk around the strong coupling point.
• For VgtgtTK

• Numerically Non crossing approximation (NCA) ?
  A self consistent perturbation theory around the
  atomic limit.
• Analytically Scaling argument proposed by
  Kaminsky Glazman 01 ? A rescaling of
  perturbation theory in the exchange coupling.

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Schemes for Calculation of t?fx(0)

• RPT for VltltTK
• Fermi liquid fixed point ? Anderson model with
  renormalized parameters and counter terms.
• Perturbation theory in e/TK,T/TK and V/TK

The spatial dependence enters through the
polynomials
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Schemes for Calculation of t?fx(0)

• NCA scheme for VgtgtTK
• Spectral function Ad, and FdGdlt/(2piAd).
• ? I(e)2ci?-1GAd(e)Fd(e)-f(0)(e).
  
  
• Acounts for inelastic scatterings to produce
  Fd?f(0).
• Not fit for VltltTK does not reproduce FL.

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Schemes for Calculation of t?fx(0)

• Scaling Scheme for VgtgtTK
• PT in ?J and rescaling.
• PT collision
  integral kernel
• The Koringa rate

• Rescaling

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Breakdown of perturbation theory in ci

• Small parameter of ci expansion
• Maximal at crossover

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Future direction
.
dense Heavy Fermion wires

• Micron sized filaments are formed in the melt of
  the heavy Fermion alloy UPt3.
• Is it possible to realize a non equilibrium
  distribution is such a wire?
• If so how would the non-equilibrium state effect
  the competition between RKKY mediated magnetism
  Kondo induced heavy fermions formation?

T
AF
PM
J
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Conclusion

• The shot-noise and distribution function in DC
  biased diffusive meso-wires hosting magnetic
  impurities were studied.
• In the dilute limit the impurities are
  equivalent to DC biased quantum dots.
• Low frequency shot-noise is an ideal probe of
  inelastic scatterings in this non-equilibrium
  Kondo system.
• The inelastic scattering rate is enhanced in the
  crossover reflected in a noise peak in VTK.