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

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:

random potential random phases only constructive interference of time-reversed pathes

weak localization (determined by return probabílity)

classical quantum

interference correction to conductivity:

return probability due to diffusion

Weak localization in weakly disordered metal

Interference:

random potential random phases only constructive interference of time-reversed pathes

weak localization (determined by return probabílity)

classical quantum

interference correction to conductivity:

loss of coherence after timedue to dephasing

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 fluxmeasure change in resistivity

flux quantum enclosed after time

Mohanty, Jariwala, Webb (1996)

Saturation of dephasing rate at T=0?

Extrinsic origin of residual dephasing? heating, external noise etc. experimentally excludedIntrinsic 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

Dephasing at T=0?

typical sizes of wires:50nm x 100nm x 300m

Pierre,Pothier et al. (03)Ag, Cu, Au wires5N = 99.999%6N = 99.9999%

extremely clean wiresfollow Altshuler, Aronov,Khmelnitzkii (82) prediction for e-e interactions

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

model and diagrams

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

model and diagrams

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)

Mohanty et al. 1996 Schopfer, Bäuerle et al. (03)15 ppm iron in gold

Doping by magnetic Fe impurities

approx. constant dephasing rate forapprox. linear rate for

goal: calculate exact dephasing rateno fit parameters if concentration and (and ) known

Is random for large ?

from 1-loop RG

randomness from short-range physicsposition of magnetic impurity in unit cell, clustering of impurities etc.

may or may not be present

randomness from long-range physics:

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

Weak localization and Kondo:self energy and vertex correction for

self energy given by T-matrix:

two typesof vertices:

Weak localization and Kondo:self energy and vertices of Cooperon for

self energy given by T-matrix:

two typesof vertices:

include in first step only self-energies and elastic vertex corrections: neglect inelastic vertexlater: exact for small density

solution of Bethe-Salpeter equation simpleas inelastic vertex neglected:

total cross-section elastic cross-section

inelastic cross-section

in Anderson impurity model with hybridization

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 : FM clusters of two spins make spin-glass with

All corrections negligible for experimentally relevant parameters!

Results: What is ?

• both and T dependence of important define -independent with same WL correction

• dependence on dimension and B accidentally smalle.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

Bäuerle et al., preprint (06) solid curves: NRG for S=1/2 (blue), S=1 (red), S=3/2 (green)

• 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?

similar: Alzoubi, Birge, preprint (06)

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

Pierre and Bierge (02)

Suppression of Kondo dephasing by magnetic fieldstudy Aharonov-Bohm oscillations

Aharonov Bohm: periodic signal on top of UCFs

Theory: dephasing of Aharonov-Bohm oszillations

Conductance fluctuations periodic in flux quantum:

What is relevant energy?

(exponentially rare high-energy excitations may dominate due to smaller dephasing)

Experimentally: limit irrelevant but some dependence on

(for d=1, more complicated in d>1, 2 frequencies)

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 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)

NRG (Costi)

Resistivity (Mallet et al preprint 06)

Origin of saturation of dephasing rate?

But unclear: What are relevant impurities?Role of larger spin?Distribution of spin-orbit coupling?

Easily fitted by some distribution of magn. impurities