Current noise in 1D electron systems ISSP International Summer School August 2003

1. Current noise in 1D electron systemsISSP International Summer SchoolAugust 2003 Björn Trauzettel Albert-Ludwigs-Universität Freiburg, Germany [Tans et al., Nature 1997] [Chung et al., PRB 2003]

2. Why is it interesting? [de-Picciotto et al., Nature 389, 162 (1997)] [Saminadayar et al., PRL 79, 2526 (1997)] direct observation of fractional charge ?!

3. Important questions: • Is it possible to measure a fractional charge in two terminal shot noise experiments on carbon nanotubes? • Can we understand the experiments by de-Picciotto et al. and Saminadayar et al. in terms of the Tomonaga-Luttinger-Liquid (TLL) model?

4. 1. Part:Interpretation of shot noise experiments on FQH edge state devices

5. Reminder of TLL model Low energy fixed point Hamiltonian: interaction parameter: Electron field operator (in bosonization): Klein factors

6. Impurity in a TLL can be scaled away by a unitary transformation dominant contribution at low energies Fixed point Hamiltonian: • corresponds to tunneling of quasiparticles with charge e*=eg • bears a resemblance to the boundary sine-Gordon Hamiltonian

7. Coupling of external voltage • fundamental difference between a chiraland a non-chiral TLL system • chiral TLL system voltage drop approach • non-chiral TLL system  different methods (e.g. the g(x) model, etc.) yield the conductance (in contrast to the experimental observation by Tarucha, Honda, and Saku, SSC 94, 413 (1995)) [derived by: Maslov and Stone, Ponomarenko, Safi and Schulz, Kawabata, Shimizu, etc., using different methods and ways of thinking about the problem]

8. Shot noise Perturbative calculations in Keldysh formalism give: strong backscattering limit weak backscattering limit [Kane and Fisher, PRL 72, 724 (1994)]

9. Strategy for non-perturbative calculation • find the appropriate excitations of the boundary sine-Gordon model (kink, anti-kink, breathers) • particles are almost free with a kind offractional statistics that depend on the energy and the interactions ( TBA equations) • local operators act in a quite complicated fashion on the quasi particle basis • however, the total charge operatoracts diagonally on this basis  calculation of the current and the noise is not so messy • apply the Landauer-Büttiker formalism to these particles [Fendley, Ludwig, and Saleur, PRL + PRB (1995-96)]

10. Exact solution for g=1/2 Expression for the shot noise at finite temperature: with the effective transmission coefficient The right(+) and left(-) moving quasiparticles obey the distribution function [Fendley and Saleur, PRB 54, 10845 (1996)]

11. Heuristic formulas for the noise Simple IPM: constant transmission Advanced IPM: with [used to interpret the data of: de-Picciotto et al., Nature 389, 162 (1997); Reznikov et al., Nature 399, 238 (1999); Griffiths et al., PRL 85, 3918 (2000).]

12. Comparison of heuristic formulas and exact solution for the case g=1/2 weak backscattering limit (t=0.95) strong backscattering limit (t=0.14) [Glattli, Roche, Saleur, and Trauzettel, in preparation]

13. 2. Part:Shot noise of non-chiral TLL systems (i.e. carbon nanotubes, cleaved edge overgrowth quantum wires, etc.)

14. Physical system • has to take into account the non-interacting nature • of the Fermi liquid leads • one way to consider this: g(x) step function model shifts band bottom in leads  electroneutrality [Maslov and Stone; Ponomarenko; Safi and Schulz, PRB (1995); Furusaki and Nagaosa, PRB (1996)]

15. Inhomogeneous correlation function equations of motion: • find the eigenfunctions of the inhomogeneous Laplacian Special situation x=y: UV cutoff

16. Calculation of the current Current (in bosonization): particular solution of the motion determined by the full action (based on radiative boundary condition approach) • obtain the four-terminal voltage drop V(U) by requiring that [see e.g., Egger and Grabert, PRL 77, 538 (1996); 80, 2255(E)]]

17. Results for the backscattered current order 2 calculation [Dolcini, Grabert, Safi, and Trauzettel, in preparation]

18. Calculation of the true shot noise path integral with respect to the full action evaluation of the path integral at order 2 yields • no visibility of fractional charge in the weak • backscattering limit • valid for any interaction strength g • due to the assumption that  < vF/L [Ponomarenko and Nagaosa, PRB (1999); Trauzettel, Egger, and Grabert, PRL (2002)]

19. What happens at higher frequencies? • We still talk about shot noise at zero temperature, but we look at two regimes: L<<<<eU and <<L<<eU with L=vF/gL. • Finite frequency excess noise: •  << L   = 1 : •  >> L  <> = g : • at high frequencies and/or for long quantum wires, it should indeed be possible to observe a fractional charge

20. Experimental situation: non-chiral TLLs [Roche et al., EPJB 28, 217 (2002)] • shot noise experiments on CNT ropes • very good contacts, no dominant backscatterer • extreme low Fano factor (lower than 1/100)

21. Summary and open questions • Experimental observations of fractional charge in FQH devices can be understood within the TLL model • Fractional charge might be visible in non-chiral realizations of TLLs at sufficiently high frequencies • Interesting aspects of finite frequency noise? • Role of less relevant impurity operators for the interpretation of noise experiments? [see e.g., Chung et al., PRB 67, R201104 (2003) and Koutouza, Saleur, and Trauzettel, PRL 2003]

22. In collaboration with: Christian Glattli (CEA Saclay, France) Patrice Roche (CEA Saclay, France) Hubert Saleur (CEA Saclay, France) Fabrizio Dolcini (Freiburg, Germany) Reinhold Egger (Düsseldorf, Germany) Hermann Grabert (Freiburg, Germany) Inès Safi (Orsay, France)