Transcript Folie 1

Dephasing by magnetic impurities
Tobias Micklitz, A. Altland and A. Rosch, University of Cologne
T. A. Costi, FZ Jülich
• what is dephasing?
• dephasing and weak localization
• exact, universal dephasing rate due to
diluted Kondo impurities
What is dephasing?
• depends on whom you ask and
on 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 matrix
e.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
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
Weak localization in weakly disordered metal
Interference:
classical
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
quantum
Origins of dephasing
Pothier
• electron – phonon interactions
• electron – electron interactions
• interactions with dynamical impurities
(magnetic impurities, two-level systems…)
Measuring dephasing rates:
idea: destroy interference of time-reversed pathes by
magnetic flux
measure change in resistivity
F
flux quantum enclosed after time
Saturation of dephasing rate at T=0?
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 level
but: only perturbative results known
Dephasing at T=0?
extremely clean wires
follow Altshuler, Aronov,
Khmelnitzkii (82) prediction
for e-e interactions
typical sizes of wires:
50nm x 100nm x 300mm
Pierre,Pothier et al. (03)
Ag, Cu, Au wires
5N = 99.999%
6N = 99.9999%
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
model and diagrams
• model: weakly disordered metal
plus diluted spin-1/2 Kondo impurities
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
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)
Doping by magnetic Fe impurities
Mohanty et al. 1996
Schopfer, Bäuerle et al. (03)
15 ppm iron in gold
approx. constant dephasing rate for
approx. linear rate for
goal: calculate exact dephasing rate
no fit parameters if concentration and
(and
) known
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
Result: fluctuations of
can be neglected for
(rare regions: exponentially small contribution to dephasing rate)
diagrammatically:
neglect mixed Kondo/disorder diagrams
technically: suppressed as
large
however: can become important at low T (later)
Disorder and interactions well separated
Weak localization and Kondo:
self energy and vertex correction for
self energy given by T-matrix:
two types
of vertices:
Weak localization and Kondo:
self 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
solution of Bethe-Salpeter equation simple
as 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)
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….
Corrections 2:
weak localization correction to dephasing rate
always suppressed by large
but wins at low T in d<2:
only relevant in 1d for
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
make spin-glass with
: FM clusters of two spins
All corrections negligible for experimentally relevant parameters!
Results: What is
?
• both e and T dependence of
define e-independent
important
with same WL correction
• dependence on dimension and B accidentally small
e.g. from Fermi liquid theory
Results: universal dephasing rate
T-matrix calculated using numerical renormalization group (T. A. Costi)
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
comparison to experiment
• to do: determine
and
independently
• here: Fe ions
successful fit to spin ½
• densities OK but factor
2 discrepancy in
• saturation
!!!
• Fe: S=2?
underscreened?
NO (compare to S=1, 3/2)
• Role of spin-orbit?
Conclusion: most Fe perfectly screened
Bäuerle et al., preprint (06)
saturation: some Fe close to other defects solid curves: NRG for S=1/2 (blue), S=1 (red)
S=3/2 (green)
or extra dynamical defects
from implantation process?
similar: Alzoubi, Birge, preprint (06)
Interplay of electron-electron interactions
and dephasing from Kondo impurities?
• Does electron-electron interaction strongly affect
Kondo-dephasing? Probably not
(small changes of energy averaging)
• Does Kondo-dephasing strongly affect electron-electron
interactions? Yes: infrared divergencies dominate
dephasing due to electron-electron interactions
in 1d:
• not additive
do not subtract background, fit instead
Suppression of Kondo dephasing by magnetic field
study Aharonov-Bohm oscillations
Pierre and Bierge (02)
Aharonov Bohm: periodic signal on top of UCFs
Theory: dephasing of Aharonov-Bohm oszillations
Conductance fluctuations periodic in flux quantum:
(for d=1, more complicated in d>1, 2 frequencies)
What is relevant energy?
(exponentially rare high-energy excitations may dominate
due to smaller dephasing)
Experimentally: limit irrelevant but some dependence on
Results: effective dephasing rate:
dependence on Zeeman field
L=10 Lhit
Conclusions:
• for diluted dynamical impurities: dephasing-rate determined by
inelastic scattering cross-section
• universal dephasing rate easily calculable
• presently no experiments on spin ½ impurities
but 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 nanodomains, 2-channel Kondo: vertex corrections important
• microscopics of saturation of dephasing rate?
T. Micklitz, A. Altland, T. A. Costi, A. Rosch, PRL (2006)
NRG (Costi)
Resistivity (Mallet et al preprint 06)
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?