Non equilibrium noise as a probe of the Kondo effect in mesoscopic wires

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

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

5. What happens when the bias is reduced to the Kondo temperature and below,V~TK and V<<Tkwhere 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. 0<x<1 distance from the left electrode divided by length of wire L Mesoscopic wire Elastic distribution function Macroscopic wire Local Equilibrium distribution [ Nagaev, 92]: Noise probes inelastic processes.

8. How does in-1 dueto magnetic impurity mediated interaction depends on bias? Small Bias: V<<TK 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 V2.

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

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

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 nin<<1: perturbation theory in ci. t≶[fx(0)]equivalent to quantum dot t-matrices. Coupled by and to electrodes at chemical potentials μL and μR. • Schematically: • nin<<1 electrons do not scatter inelastically twice • Diffusion communicates the distribution from the leads

16. Schemes for Calculation of t≶[fx(0)] • Non equilibrium Kondo problem - an open problem. No reliable approach to describes the crossover regime V~TK. • Analytically for V<<TK: extension of Hewson’s renormalized perturbation theory (RPT) to weak non-equilibrium → Perturbation theory in ε/TK, T/TK and V/Tk around the strong coupling point. • For V>>TK • 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≶[fx(0)] • RPT for V<<TK: • Fermi liquid fixed point → Anderson model with renormalized parameters and counter terms. • Perturbation theory in ε/TK,T/TK and V/TK : The spatial dependence enters through the polynomials:

18. Schemes for Calculation of t≶[fx(0)] • NCA scheme for V>>TK • Spectral function Ad, and Fd=Gd</(2πiAd). • ⇒I(ε)=2ciρ-1ΓAd(ε)[Fd(ε)-f(0)(ε)]. • Acounts for inelastic scatterings to produce Fd≠f(0). • Not fit for V<<TK: does not reproduce FL.

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

22. Breakdown of perturbation theory in ci • Small parameter of ci expansion – • Maximal at crossover

23. Future direction: . dense Heavy Fermion wires • Micron sized filaments are formed in the melt of the heavy Fermion alloy UPt3. • 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 inducedheavy fermions formation? T AF PM J

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~TK.