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**Interactions between electrons, mesoscopic Josephson effect and **

asymmetric current fluctuations B. Huard & Quantronics group

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**important for L < Lj : phase coherence length**

Quantum electronics Macroscopic conductors 2 I I DC AMPS DC AMPS L L/2 R L Mesoscopic conductors R L Quantum mechanics changes the rules important for L < Lj : phase coherence length

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**Overview of the thesis d Tool for measuring**

1) Phase coherence and interactions between electrons in a disordered metal 150 nm 2) Mesoscopic Josephson effects ) Measuring high order current noise superconductor V B I I t d Tool for measuring the asymmetry of I(t) ? I(d) for elementary conductor

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**Overview of the thesis d Tool for measuring**

1) Phase coherence and interactions between electrons in a disordered metal 150 nm 2) Mesoscopic Josephson effects ) Measuring high order current noise superconductor V B I I t d Tool for measuring the asymmetry of I(t) ? I(d) for elementary conductor

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**Electron dynamics in metallic thin films**

+ le 150 nm Grain boundaries Film edges Impurities Elastic scattering - Diffusion - Limit conductance Inelastic scattering Coulomb interaction Phonons Magnetic moments - Limit coherence (Lj) - Exchange energy Typically, lF le Lj ≤ L

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**How to access e-e interactions ?**

1st method : weak localization R(B) measures Lj B In a wire Pierre et al., PRB (2003) B (mT) First measurement: Wind et al. (1986)

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**How to access e-e interactions ?**

2nd method : energy relaxation U Diffusion time : (20 ns for 20 µm) Occupied states E ? eU U=0 f(E)

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**Distribution function and**

energy exchange rates « weak interactions » U tD tint. E eU f(E)

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**Distribution function and**

energy exchange rates « strong interactions » U tD tint. E eU f(E)

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**Distribution function and**

energy exchange rates « weak interactions » « strong interactions » tD tint. tD tint. E E f(E) f(E) f(E) interactions

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**Understanding of inelastic scattering**

1st method Weak localization 2nd method Energy relaxation Interaction stronger than expected OK Coulomb interaction Wind et al. (1986) Pierre et al. (2000) e (µeV) Probed energies : 0.01 0.1 1 10 100 dependence on B as expected Magnetic moments OK Pierre et al. (2003) Anthore et al. (2003)

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**several explanations dismissed Quantitative experiment**

Understanding of inelastic scattering 1st method Weak localization 2nd method Energy relaxation Interaction stronger than expected OK Coulomb interaction Wind et al. (1986) Pierre et al. (2000) dependence on B as expected Magnetic moments OK Pierre et al. (2003) Anthore et al. (2003) several explanations dismissed (Huard et al., Sol. State Comm. 2004) Quantitative experiment (Huard et al., PRL 2005)

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**Access e-e interactions : measurement of f(E)**

Dynamical Coulomb blockade (ZBA) R I U=0 mV

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**Measurement of f(E) Dynamical Coulomb blockade (ZBA) weak interaction**

strong interaction U=0.2 mV U=0 mV

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**Quantitative investigation of the effects of magnetic impurities**

0.65 ppm Mn implantation implanted bare Ag ( %) Left as is Comparative experiments using methods 1 and 2 Huard et al., PRL 2005

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**1st method : weak localization**

spin-flip Coulomb phonons 0.65 ppm Mn 0.65 ppm consistent with implantation 0.03 ppm compatible with < 1ppm dirt Best fit of Lj(T) for

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**2nd method : energy relaxation**

implanted 0.65 ppm Mn strong interaction U = 0.1 mV B = 0.3 T T= 20 mK bare weak interaction

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*** Spin-flip scattering on a magnetic impurity energy E E f(E)**

- dephasing - no change of energy At B=0 energy E E f(E) E E * rate maximal at Kondo temperature

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*** Interaction between electrons mediated by a magnetic impurity**

Virtual state E E-e E’+e E’ f(E) E E-e E’ E’+e * Enhanced by Kondo effect Kaminski and Glazman, PRL (2001)

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*** Interaction mediated by a magnetic impurity :**

effect of a low magnetic field (gµBeU) Virtual state E E-e E’+e E-EZ EZ=gµB E’ f(E) E E-e E’ E’+e * Modified rate (e-EZ)-2

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**Spin-flip scattering on a magnetic impurity : **

effect of a high magnetic field (gµB eU) Virtual state E eU EZ E-e E’+e E’ E-EZ f(E) Reduction of the energy exchange rate Modified rate (e-EZ)-2

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**Experimental data at low and at high B**

implanted 0.65 ppm Mn U = 0.1 mV B = 0.3 T (gµBB = 0.35 eU) B = 2.1 T (gµBB = 2.4 eU) Very weak interaction bare U = 0.1 mV T= 20 mK

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Various B and U T= 20 mK

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**Comparison with theory**

Using theory of Goeppert, Galperin, Altshuler and Grabert PRB (2001) Only 1 fit parameter for all curves : ke-e=0.05 ns-1.meV-1/2 (Coulomb interaction intensity)

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**Unexplained discrepancy**

Coulomb interaction intensity ke-e Experiments on 15 different wires: e (µeV) 1 ) -1/2 100 meV -1 10 0.1 1 best fit for ke-e (ns energy relaxation weak localization 0.1 0.01 0.02 0.02 0.1 1 expected for ke-e (ns -1 meV -1/2 ) Unexplained discrepancy

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**Conclusions on interactions**

Quantitative understanding of the role played by magnetic impurities but Coulomb interaction stronger than expected Coulomb spin-flip phonons

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**Overview of the thesis d Tool for measuring**

1) Phase coherence and interactions between electrons in a disordered metal 150 nm 2) Mesoscopic Josephson effects ) Measuring high order current noise superconductor V B I I t d Tool for measuring the asymmetry of I(t) ? I(d) for elementary conductor

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**Unified theory of the Josephson effect**

Case of superconducting electrodes B I Supercurrent through a weak link ? Unified theory of the Josephson effect Furusaki et al. PRL 1991, …

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**Transmission probability**

Conduction channels Coherent Conductor (L«Lj) V I Landauer Collection of independent channels r r’ t t’ Transmission probability

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**probability amplitude**

Andreev reflection (1964) N S "e" "h" a(E)e-if "e" "h" a(E)eif a(E)e-if Andreev reflection probability amplitude

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**Andreev bound states t = 1 fR fL 2 current carrying bound states**

in a short ballistic channel ( < x ) t = 1 fL fR a(E)eif "e" a(E)e-if L R "h" "h" "e" E(d) 2 current carrying bound states +D E→ d p 2p E← -D ← →

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**Andreev bound states t < 1**

in a short ballistic channel ( < x ) t < 1 fL fR a(E, fL) a(E, fR) "e" "h" "h" E(d) +D E+ d p 2p Central prediction of the mesoscopic theory of the Josephson effect -D E- A. Furusaki, M. Tsukada (1991)

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**Andreev bound states t < 1 d I(d,t)**

in a short ballistic channel ( < x ) t < 1 fL fR a(E, fL) a(E, fR) "e" "h" "h" CURRENT I(d,t) E(d) d p 2p d 2p Central prediction of the mesoscopic theory of the Josephson effect -D A. Furusaki, M. Tsukada (1991)

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**Quantitative test using atomic contacts .**

Atomic orbitals I V S S { t1 … tN } A few independent conduction channels of measurable and tunable transmissions J.C. Cuevas et al. (1998) E. Scheer et al. (1998) I-V { t1 … tN } Quantitative test

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**Atomic contact pushing rods sample counter-support with shielded coil**

metallic film pushing rods Flexible substrate insulating layer counter- support counter-support with shielded coil 3 cm

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**How to test I(d) theory It Strategy : Measure {t1,…,tM} Measure I(d)**

V Tunnel junction j Al It Metallic bridge (atomic contact) Ib Strategy : Measure {t1,…,tM} Measure I(d) V>0 V=0

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**Switching of a tunnel junction .**

Ib It V Ib (circuit breaker) I open circuit : 2D/e >V>0 2D/e V circuit breaker : Ib>I V>0 stable

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**Measure {t1,…,tM} It V Measure I(V) Ib Transmissions AC3 0.992 ± 0.003**

method: Scheer et al. 1997 Transmissions Measure I(V) It AC3 0.992 ± 0.003 0.089 ±0.06 0.088 ±0.06 Ib AC2 0.957 ± 0.01 0.185 ±0.05 AC1 0.62 ± 0.01 0.12 ±0.015 0.115 ±0.01 0.11 ±0.01 0.11 ±0.01

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**Measure I(d) It d j /f0 + p/2 V Ib I Ibare Ib j (circuit breaker)**

2D/e V d j /f0 + p/2

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Measure I(d) 0.62 ± 0.01 0.957 ± 0.01 0.992 ± 0.003

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**Comparison with theory I(d)**

Theory : I(d) + switching at T0 0.62 ± 0.01 0.957 ± 0.01 0.992 ± 0.003

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**Overall good agreement**

Comparison with theory I(d) Theory : I(d) + switching at T0 Overall good agreement but with a slight deviation at t 1 0.62 ± 0.01 0.957 ± 0.01 0.992 ± 0.003

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**Overview of the thesis d Tool for measuring**

1) Phase coherence and interactions between electrons in a disordered metal 150 nm 2) Mesoscopic Josephson effects ) Measuring high order current noise superconductor V B I I t d Tool for measuring the asymmetry of I(t) ? I(d) for elementary conductor

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**Full counting statistics**

n Vm t Average current during t ne/t=It Pt(n) characterizes It pioneer: Levitov et al. (1993) Need a new tool to measure it t

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**Well known case : tunnel junction**

Independent tunnel events Poisson distribution n Log scale Pt(n) n Pt(n) is asymmetric Simple distribution detector calibration

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Which charge counter ? Tunnel junction Vm It It t

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**Charge counter: Josephson junction**

Clarge dIm RlargeClarge 20 µs Vm Im Rlarge Im I G+ Im Switching rates G- -I t Proposal : Tobiska & Nazarov PRL (2004)

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**Charge counter: Josephson junction**

dIm Ib dIm+Ib Ib Vm Im G+ dIm -Ib G- dIm +Ib I I Im -I -I t t

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**Asymmetric current fluctuations**

Ib (µA) so that G G 30 kHz Im (µA)

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**Asymmetric current fluctuations**

G+/ G- -1 |Ib| so that G+ cste (30 kHz) Gaussian noise Im (µA) There is an asymmetry

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**Asymmetric current fluctuations**

G+/ G- -1 |Ib| so that G+ cste (30 kHz) Ankerhold (2006) Im (µA) Disagreement with existing theory

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**Conclusions Decoherence and interactions in Quantitative experiments**

disordered metals Quantitative experiments Open : Coulomb intensity Quantitative agreement with fundamental relation Persp. : spectro and manip. of Andreev states Unified theory of Josephson effect I (nA) j Tool for measuring high order current noise Tool sensitive to high order noise OK Open : Interpretation ?

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**Coulomb interaction discrepancy explanations**

Extrinsic energy exchange processes ? Quasi-1D model inappropriate ? Diffusive approximation invalid ? Hartree term stronger than expected ? Theory valid at equilibrium only ? Magnetic impurities and 2 level systems cannot explain the discrepancy (bad fits) Slight error at the lowest probed energies would furthermore reduce the intensity ke-e Never been investigated Strong enough if Ag very close to ferromagnetic instability Yes, same result close to equilibrium f(E) 1 Experiment near equilibrium E

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