what is dephasing? dephasing and weak localization

1. 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

2. 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 experimentclose to equilibrium, expect: no dephasing at

3. 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

4. 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

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

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

7. Saturation of dephasing rate at T=0? Mohanty, Jariwala, Webb (1996) Extrinsicorigin of residual dephasing? heating, external noise etc. experimentally excluded Intrinsicorigin? 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

8. Dephasing at T=0? extremely clean wiresfollow 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%

9. 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

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

11. 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

12. 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)

13. 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

14. 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

15. Result: fluctuations of can be neglected for (rare regions: exponentially small contribution to dephasing rate)diagrammatically:neglect mixed Kondo/disorder diagramstechnically: suppressed as largehowever: can become important at low T (later) Disorder and interactions well separated

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

17. 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 elasticvertex corrections: neglect inelasticvertex later: exact for small density

18. 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)

19. 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….

20. 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

21. 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!

22. 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

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

24. 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

25. comparison to experiment • to do: determineand independently • here: Fe ionssuccessful fit to spin ½ • densities OK but factor2 discrepancy in • saturation !!! • Fe: S=2? underscreened?NO (compare to S=1, 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 S=1/2 (blue), S=1 (red), S=3/2 (green) similar: Alzoubi, Birge, preprint (06)

26. 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

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

28. 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

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

30. 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)

31. NRG (Costi)

32. Resistivity (Mallet et al preprint 06)

33. 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?