Dephasing by magnetic impurities Tobias Micklitz A Altland

Dephasing by magnetic impurities Tobias Micklitz, A. Altland 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: 50 nm x 100 nm x 300 mm Pierre, Pothier et al. (03) Ag, Cu, Au wires 5 N = 99. 999% 6 N = 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 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 1 d for

Corrections 3: Altshuler Aronov • lowest T: non-local interaction effects get important (same universality class as disordered Fermi liquid) e. g. in 2 d (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 1 d: • 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?