1. Non equilibrium noise as a probe of the Kondo effect in mesoscopic wires Eran Lebanon Rutgers University with Piers Coleman arXiv: cond-mat/0501001 DOE grant DE-FE02-00ER45790

2. Outline Motivation: relevance of magnetic impurities in diffusive wires Enhanced inelastic scattering at and noise peak PT in impurity concentration; equivalence to a quantum dot Calculation schemes: -RPT -NCA, coupling scaling. Break down of the perturbation theory.

3. Diffusive metallic wire SC V s-w V Anomalous collision integral Kernel!

4. Quantitative comparisons to experimental data: Kroha & Zawadowski PRL 88, 176803 (2002) Göppert & Grabert PRB 64, 033301 (2001) Consistent with later experimental observation: Energy relaxation is stronger for the less pure metals The energy relaxation is quenched by a magnetic field

5. What happens when the bias is reduced to the Kondo temperature and below, V~T K and V<<T k where Kondo correlations become strong? Experiment+theory: In gold and copper wires the energy relaxation is dominated by magnetic impurity mediated interaction and not by direct electron-electron interaction even for dilute doping (~1ppm)

6. Thouless Energy Energy relaxation Rate Macroscopic wire: Local Equilibrium Mesoscopic wire: Elastic distribution

7. Mesoscopic wire Elastic distribution function 0<x<1 distance from the left electrode divided by length of wire L Macroscopic wire Local Equilibrium distribution [ Nagaev, 92]: Noise probes inelastic processes.

8. How does  in -1 due to magnetic impurity mediated interaction depends on bias? Small Bias: V<<T K At T=0, V=0 : The impurity’s magnetic moment is screened completely and the Kondo singlet is fully developed. The physics is described by a local Fermi liquid fixed point: there are no inelastic scatterings at low energies. For small V: The energy relaxation increases with the bias and is proportional to V 2.

9. How does  in -1 due to magnetic impurity mediated interaction depends on bias? Large Bias: V>>T K The infra red singularities of perturbation theory are cutoff by the bias. The problem becomes a weak coupling problem with an effective coupling J eff ~ ln -1 ( eV / k B T K ). The relaxation rate decays with V like a polynomial of J eff = ln -1 ( eV / k B T K ).

10. Enhancement of the inelastic scattering rate for intermediate biases V~T K. This will be manifested in a peak in both the noise curve and the intensity of the distribution smearing as functions of the bias.  in -1 : ↗ ↘ V >T K Non equilibrium reminiscent of the equilibrium dephasing rate peak

11. Model Operators: creates an electron on i-th impurity, occupation of i- th impurity, conduction electron field operator. Parameters: impurity orbital energy, on-site repulsion, mixing amplitude. LMR:

12. Model creates electron on i-th impurity, occupation of i-th impurity, conduction electron field operator. impurity orbital energy, on-site repulsion, mixing amplitude. LMR:

13. Boltzmann equation Baym Kadanoff Fourier transform with respect to relative coordinates Gradient expansion keeping Summation over the momentum The current in the system is diffusive For dilute magnetic impurities:

14. Perturbation theory in the impurity concentration without impurities: The cores. noise:

15. Equivalence to a quantum dot problem For n in <<1: perturbation theory in c i. t ≶ [f x (0) ] equivalent to quantum dot t-matrices. Coupled by and to electrodes at chemical potentials μ L and μ R. Schematically: n in <<1 electrons do not scatter inelastically twice Diffusion communicates the distribution from the leads

16. Schemes for Calculation of t ≶ [f x (0) ] Non equilibrium Kondo problem - an open problem. No reliable approach to describes the crossover regime V~T K. Analytically for V<<T K : extension of Hewson’s renormalized perturbation theory (RPT) to weak non-equilibrium → Perturbation theory in ε/T K, T/T K and V/T k around the strong coupling point. For V>>T K  Numerically: Non crossing approximation (NCA) → A self consistent perturbation theory around the atomic limit.  Analytically: Scaling argument proposed by [Kaminsky & Glazman 01] → A rescaling of perturbation theory in the exchange coupling.

17. Schemes for Calculation of t ≶ [f x (0) ] RPT for V<<T K : Fermi liquid fixed point → Anderson model with renormalized parameters and counter terms. Perturbation theory in ε/T K,T/T K and V/T K : The spatial dependence enters through the polynomials:

18. Schemes for Calculation of t ≶ [f x (0) ] NCA scheme for V>>T K Spectral function A d, and F d =G d < /(2πiA d ). ⇒ I(ε)=2c i ρ -1 ΓA d (ε)[F d (ε)-f (0) (ε)]. Acounts for inelastic scatterings to produce F d ≠f (0). Not fit for V<<T K : does not reproduce FL.

19. Schemes for Calculation of t ≶ [f x (0) ] Scaling Scheme for V>>T K : PT in ρJ and rescaling. PT: collision integral kernel The Koringa rate Rescaling

22. Breakdown of perturbation theory in c i Small parameter of c i expansion – Maximal at crossover

23. Future direction:. dense Heavy Fermion wires Micron sized filaments are formed in the melt of the heavy Fermion alloy UPt 3. Is it possible to realize a non equilibrium distribution is such a wire? If so how would the non-equilibrium state effect the competition between RKKY mediated magnetism – Kondo induced heavy fermions formation? T J AF PM

24. Conclusion The shot-noise and distribution function in DC biased diffusive meso-wires hosting magnetic impurities were studied. In the dilute limit: the impurities are equivalent to DC biased quantum dots. Low frequency shot-noise is an ideal probe of inelastic scatterings in this non-equilibrium Kondo system. The inelastic scattering rate is enhanced in the crossover – reflected in a noise peak in V~T K.