Dephasing by magnetic impurities Tobias Micklitz , A. Altland and A. - - PowerPoint PPT Presentation

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 dephasing rate due to

diluted Kondo impurities

What is dephasing?

• depends on whom you ask and
• n 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

• nly constructive interference
• f time-reversed pathes

weak localization

(determined by return probabílity)

classical quantum interference correction to conductivity: loss of coherence after time due to dephasing

Origins of dephasing

Pothie r

• 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

flux quantum enclosed after time

Φ

Mohanty, Jariwala, Webb (1996)

Saturation of dephasing rate at T=0?

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?

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

Pierre,Pothier et al. (03) Ag, Cu, Au wires 5N = 99.999% 6N = 99.9999%

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

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

Implanting magnetic Fe impurities

• approx. constant dephasing rate for
• approx. linear rate for

goal: calculate exact dephasing rate no fit parameters if concentration and known

Is random for large ?

from 1-loop RG 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:

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 vertices for self energy given by T-matrix: two types

• f 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 Δ

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:

Corrections 2: weak localization correction to dephasing rate

always suppressed by large but wins at low T in d<2:

• nly 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 below extremely low crossover-temperature

All corrections negligible for experimentally relevant parameters!

Results: What is ?

• both ε and T dependence of important

define ε-independent 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

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

• Proper comparison to experiment:

not yet done

• to do: determine

and independently

• here: naïve fit works

much too well (background!!)

• role of spin S>1/2 and

spin-orbit coupling?

theory

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

Conclusions:

• for diluted dynamical impurities: dephasing-rate

determined by inelastic scattering cross-section

• universal dephasing rate easily calculable

Outlook:

• comparison to experiment: without fitting parameter
• Aharonov-Bohm oscillations (magn. fields), universal

conductance fluctuations, persistent currents, ….

• ferromagnetic impurities, larger spins, fluctuating nano-

domains, 2-channel Kondo: vertex corrections important

• T. Micklitz, A. Altland, T. A. Costi, A. Rosch, cond-mat/0509583