1. Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A
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

2. correlated
electron
systems
mesoscopic physics
nano-scale phase separation (manganites, high-temperature superconductors,….)
single-electron transistors, non-equilibrium …
much simpler problem: how do interactions affect mesoscopic phenomena dephasing

3. 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

4. Weak localization in weakly disordered metal
Interference:
classical
quantum
random potential random phases
most interference terms cancel

5. 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

6. 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

7. Origins of dephasing electron – phonon interactions
Pothier
electron – phonon interactions
electron – electron interactions
interactions with dynamical impurities (magnetic impurities, two-level systems…)

8. Measuring dephasing rates:
idea: destroy interference of time-reversed pathes by
magnetic flux
measure change in resistivity
flux quantum enclosed after time F

9. 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

10. Dephasing at T=0? Pierre,Pothier et al. (03) Ag, Cu, Au wires
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 = %
6N = %

11. 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

12. model and diagrams
model: weakly disordered metal plus diluted spin-1/2 Kondo impurities

13. model and diagrams
model: weakly disordered metal plus diluted spin-1/2 Kondo impurities
exchange coupling J of magnetic impurities (e.g. Fe, Mn) to spin of conduction electrons

14. 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

15. 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)

16. Doping by magnetic Fe impurities
Schopfer, Bäuerle et al. (03)
15 ppm iron in gold
Mohanty et al. 1996
approx. constant dephasing rate for approx. linear rate for goal: calculate exact dephasing rate no fit parameters if concentration and (and ) known

17. 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

18. 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

19. Weak localization and Kondo: self energy and vertex correction for
self energy given by T-matrix:
two types
of vertices:

20. 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

21. 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)

22. 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….

23. 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

24. 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!

25. Results: What is ?
both e and T dependence of important define e-independent with same WL correction
dependence on dimension and B accidentally small e.g. from Fermi liquid theory

26. Results: universal dephasing rate
T-matrix calculated using numerical renormalization group (T. A. Costi)

27. 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

28. 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 saturation: some Fe close to other defects
or extra dynamical defects from implantation process?
Bäuerle et al., preprint (06)
solid curves: NRG for S=1/2 (blue), S=1 (red), S=3/2 (green)
similar: Alzoubi, Birge, preprint (06)

29. 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

30. Suppression of Kondo dephasing by magnetic field study Aharonov-Bohm oscillations
Pierre and Bierge (02)
Aharonov Bohm: periodic signal on top of UCFs

31. 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

32. Results: effective dephasing rate: dependence on Zeeman field
L=10 Lhit

33. Conclusions: Outlook:
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 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)

34. NRG (Costi)

35. Resistivity (Mallet et al preprint 06)

36. 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?