Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T....
-
Upload
blaise-barrett -
Category
Documents
-
view
217 -
download
0
Transcript of Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T....
![Page 1: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/1.jpg)
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
![Page 2: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/2.jpg)
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
![Page 3: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/3.jpg)
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
![Page 4: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/4.jpg)
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
![Page 5: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/5.jpg)
Origins of dephasing
Pothier
• electron – phonon interactions• electron – electron interactions• interactions with dynamical impurities
(magnetic impurities, two-level systems…)
![Page 6: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/6.jpg)
Measuring dephasing rates:idea: destroy interference of time-reversed pathes by
magnetic fluxmeasure change in resistivity
flux quantum enclosed after time
![Page 7: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/7.jpg)
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
![Page 8: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/8.jpg)
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
![Page 9: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/9.jpg)
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
![Page 10: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/10.jpg)
• model: weakly disordered metal plus diluted spin-1/2 Kondo impurities
model and diagrams
![Page 11: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/11.jpg)
• 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
![Page 12: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/12.jpg)
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)
![Page 13: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/13.jpg)
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
![Page 14: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/14.jpg)
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:
![Page 15: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/15.jpg)
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
![Page 16: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/16.jpg)
Weak localization and Kondo:self energy and vertex correction for
self energy given by T-matrix:
two typesof vertices:
![Page 17: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/17.jpg)
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
![Page 18: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/18.jpg)
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)
![Page 19: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/19.jpg)
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….
![Page 20: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/20.jpg)
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
![Page 21: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/21.jpg)
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!
![Page 22: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/22.jpg)
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
![Page 23: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/23.jpg)
Results: universal dephasing rate
T-matrix calculated using numerical renormalization group (T. A. Costi)
![Page 24: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/24.jpg)
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
![Page 25: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/25.jpg)
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)
![Page 26: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/26.jpg)
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
![Page 27: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/27.jpg)
Pierre and Bierge (02)
Suppression of Kondo dephasing by magnetic fieldstudy Aharonov-Bohm oscillations
Aharonov Bohm: periodic signal on top of UCFs
![Page 28: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/28.jpg)
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)
![Page 29: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/29.jpg)
Results: effective dephasing rate:dependence on Zeeman field
L=10 Lhit
![Page 30: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/30.jpg)
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)
![Page 31: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/31.jpg)
NRG (Costi)
![Page 32: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/32.jpg)
Resistivity (Mallet et al preprint 06)
![Page 33: Dephasing by magnetic impurities Tobias Micklitz, A. Altland and A. Rosch, University of Cologne T. A. Costi, FZ Jülich what is dephasing? dephasing and.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649ef45503460f94c073ef/html5/thumbnails/33.jpg)
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