Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T.

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Dephasing by magnetic impuritiesTobias Micklitz,
A. Altland and A. Rosch, University of
CologneT. A. Costi, FZ Jülich

• what is dephasing?
• dephasing and weak localization
• exact, universal dephasing rate due todiluted
  Kondo impurities

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mesoscopicphysics
correlated electron systems

• nano-scale phase separation (manganites,
  high-temperature superconductors,.)
• single-electron transistors, non-equilibrium
• much simpler problemhow do interactions affect
  mesoscopic phenomena dephasing

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What is dephasing?

• depends on whom you ask andon precise experiment
  
• generally loss of ability to show interference
  relevant for mesoscopics, metal-insulator
  transition, quantum computing,.
• often decay of off-diagonal elements of reduced
  density matrixe.g. dephasing of Qbit
  by coupling to bath,
  non-equilibrium experiment finite
  dephasing rate even at
• here use weak localization as interference
  experiment close to equilibrium, expect no
  dephasing at

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Weak localization in weakly disordered metal
Interference
classical
quantum
random potential random phases
most interference terms cancel
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Weak localization in weakly disordered metal
Interference
classical
quantum
random potential random phases
only constructive interference of
time-reversed pathes weak
localization (determined by return
probabílity)
interference correction to conductivity
return probability due to diffusion
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Weak localization in weakly disordered metal
Interference
classical
quantum
random potential random phases
only constructive interference of
time-reversed pathes weak
localization (determined by return
probabílity)
interference correction to conductivity
loss of coherence after time due to dephasing
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Origins of dephasing
Pothier

• electron phonon interactions
• electron electron interactions
• interactions with dynamical impurities
  (magnetic impurities, two-level systems)

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Measuring dephasing rates
idea destroy interference of time-reversed
pathes by magnetic flux measure change in
resistivity flux quantum enclosed after time
F
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Saturation of dephasing rate at T0?
Mohanty, Jariwala, Webb (1996)
Extrinsic origin of residual dephasing? heating,
external noise etc. experimentally
excluded Intrinsic origin? Dephasing by
zero-point fluctuations of EM field (Zaikin,
Golubev) theoretically excluded (Aleiner,
Altshuler, von Delft)Likely origin magnetic (or
other dynamic) impurities on ppm levelbut only
perturbative results known
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Dephasing at T0?
extremely clean wiresfollow Altshuler,
Aronov,Khmelnitzkii (82) prediction for e-e
interactions
typical sizes of wires50nm x 100nm x 300mm
Pierre,Pothier et al. (03)Ag, Cu, Au wires 5N
99.999 6N 99.9999
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Goals

• What quantity is the dephasing rate beyond
  perturbation theory?
• Is there a universal dephasing rate of magnetic
  impurities?
• Calculate it and compare to experiments!
• Study disorder strong interactions in most
  trivial limit

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model and diagrams

• model weakly disordered metal plus
  diluted spin-1/2 Kondo impurities

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model and diagrams

• model weakly disordered metal plus
  diluted spin-1/2 Kondo impurities

exchange coupling J of magneticimpurities (e.g.
Fe, Mn) tospin of conduction electrons
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model and diagrams

• model weakly disordered metal plus
  diluted spin-1/2 Kondo impurities

• Kondo effect
• interactions J grow toward low energies due to
  resonant, coherent spin-flips
• but best understood non-perturbative problem
• spin screened below Kondo temperature
• universal behavior as function of

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model and diagrams

• model weakly disordered metal plus
  diluted spin-1/2 Kondo impurities
• average over weak random nonmagnetic potential
  (Gaussian, large )
• average over positions of magnetic
  impurities,density
• interactions only due to Kondo spins (no Coulomb)

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Doping by magnetic Fe impurities
Schopfer, Bäuerle et al. (03) 15 ppm iron in gold
Mohanty et al. 1996
approx. constant dephasing rate forapprox.
linear rate forgoal calculate exact dephasing
rateno fit parameters if concentration and
(and ) known
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Is random for large ?
randomness from short-range physics position of
magnetic impurity in unit cell, clustering of
impurities etc. may or may not be
present randomness from long-range physics
from 1-loop RG
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Result fluctuations of can be neglected
for
(rare regions exponentially small contribution
to dephasing rate)diagrammaticallyneglect
mixed Kondo/disorder diagramstechnically
suppressed as largehowever can become
important at low T (later)
Disorder and interactions well separated
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Weak localization and Kondoself energy and
vertex correction for

• self energy given by T-matrix

two types of vertices
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Weak localization and Kondoself energy and
vertices of Cooperon for

• self energy given by T-matrix

two types of vertices
include in first step only self-energies and
elastic vertex corrections neglect inelastic
vertex later exact for small density
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solution of Bethe-Salpeter equation simpleas
inelastic vertex neglected
total cross-section
elastic cross-section
inelastic cross-section
in Anderson impurity model with hybridization D
inelastic cross-section, defined by Zarand,
Borda, von Delft, Andrei (04)
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Corrections 1 from inelastic vertices

• width of inelastic vertex calculation gives
• inelastic vertices negligible for
• vertex correction time reversed electrons share
  same inelastic process

relative phase typical time typical energy
transfer
Altshuler, Aronov, Khmelnitzky, Vavilov, Larkin,
Glazman.
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Corrections 2 weak localization correction to
dephasing rate

• always suppressed by large
• but wins at low T in dlt2
• only relevant in 1d for

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Corrections 3 Altshuler Aronov

• lowest T non-local interaction effects get
  important(same universality class as disordered
  Fermi liquid)
• e.g. in 2d (up to logs)
• dominates only below
• further corrections to order FM clusters
  of two spins make spin-glass with

All corrections negligible for experimentally
relevant parameters!
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Results What is ?

• both e and T dependence of
  important define e-independent
  with same WL correction
• dependence on dimension and B accidentally
  smalle.g. from Fermi liquid theory

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Results universal dephasing rate
T-matrix calculated using numerical
renormalization group (T. A. Costi)
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comparison to experiment
Mallet,Saminadayar, Bäuerle et al. preprint (06)
ion beam implantation of 0, 2.7, 27, 67 ppm Fe
in Ag
similar data Alzoubi, Birge, preprint (06)
next subtract el.-el. dephasing and rescale with
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comparison to experiment

• to do determineand independently
• here Fe ionssuccessful fit to spin ½
• densities OK but factor2 discrepancy in
• saturation !!!
• Fe S2? underscreened?NO (compare to S1, 3/2)
• Role of spin-orbit?

Conclusion most Fe perfectly screenedsaturation
some Fe close to other defects or extra
dynamical defects from implantation process?

Bäuerle et al., preprint (06) solid curves NRG
for S1/2 (blue), S1 (red), S3/2 (green)
similar Alzoubi, Birge, preprint (06)
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Interplay of electron-electron interactionsand
dephasing from Kondo impurities?

• Does electron-electron interaction strongly
  affectKondo-dephasing? Probably not (small
  changes of energy averaging)
• Does Kondo-dephasing strongly affect
  electron-electroninteractions? Yes infrared
  divergencies dominatedephasing due to
  electron-electron interactions
• in 1d
• not additive do not subtract background,
  fit instead

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Suppression of Kondo dephasing by magnetic
fieldstudy Aharonov-Bohm oscillations
Pierre and Bierge (02)
Aharonov Bohm periodic signal on top of UCFs
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Theory dephasing of Aharonov-Bohm oszillations
Conductance fluctuations periodic in flux quantum
(for d1, more complicated in dgt1, 2 frequencies)
What is relevant energy? (exponentially rare
high-energy excitations may dominate due to
smaller dephasing) Experimentally limit
irrelevant but some dependence on
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Results effective dephasing ratedependence on
Zeeman field
L10 Lhit
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Conclusions

• for diluted dynamical impurities dephasing-rate
  determined by inelastic scattering cross-section
  
• universal dephasing rate easily calculable
• presently no experiments on spin ½ impuritiesbut
  good fits to Fe ions in Ag, Au ??
• Aharonov-Bohm oscillations (magn. fields),
  universal conductance fluctuations, persistent
  currents, .

Outlook

• microscopics of Fe ions? Is saturation universal
  in experiments? Sensitivity to disorder of large
  spin/multiple channel-models?
• ferromagnetic impurities, larger spins,
  fluctuating nano-domains, 2-channel Kondo vertex
  corrections important
• microscopics of saturation of dephasing rate?

T. Micklitz, A. Altland, T. A. Costi, A. Rosch,
PRL (2006)
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NRG (Costi)
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Resistivity (Mallet et al preprint 06)
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Origin of saturation of dephasing rate?
Easily fitted by some distribution of magn.
impurities
But unclear
What are relevant impurities? Role of larger
spin?Distribution of spin-orbit coupling?