what is dephasing? dephasing and weak localization
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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. What is dephasing?. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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
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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
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Origins of dephasing
Pothier
• electron – phonon interactions• electron – electron interactions• interactions with dynamical impurities
(magnetic impurities, two-level systems…)
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Measuring dephasing rates:idea: destroy interference of time-reversed pathes by
magnetic fluxmeasure change in resistivity
flux quantum enclosed after time
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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
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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
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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
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• model: weakly disordered metal plus diluted spin-1/2 Kondo impurities
model and diagrams
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• 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
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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)
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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
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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:
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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
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Weak localization and Kondo:self energy and vertex correction for
self energy given by T-matrix:
two typesof vertices:
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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
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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)
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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….
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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
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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!
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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
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Results: universal dephasing rate
T-matrix calculated using numerical renormalization group (T. A. Costi)
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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
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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)
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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
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Pierre and Bierge (02)
Suppression of Kondo dephasing by magnetic fieldstudy Aharonov-Bohm oscillations
Aharonov Bohm: periodic signal on top of UCFs
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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)
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Results: effective dephasing rate:dependence on Zeeman field
L=10 Lhit
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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)
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NRG (Costi)
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Resistivity (Mallet et al preprint 06)
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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