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 eeg
• bears a resemblance to the boundary sine-Gordon
  Hamiltonian

7
Coupling of external voltage

• fundamental difference between a chiral and 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 of
  fractional 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 operator acts
  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 g1/2
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 g1/2
strong backscattering limit (t0.14)
weak backscattering limit (t0.95)
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 xy
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 ? lt 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
• ?Lltlt?ltlteU and ?ltlt?LltlteU with ?LvF/gL.
• Finite frequency excess noise
• ? ltlt ?L ? ? 1
• ? gtgt ?L ? lt?gt 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)