Current noise in 1D electron systemsISSP International Summer School

August 2003

Björn TrauzettelAlbert-Ludwigs-Universität Freiburg, Germany

[Chung et al., PRB 2003][Tans et al., Nature 1997]

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[de-Picciotto et al., Nature 389, 162 (1997)][Saminadayar et al., PRL 79, 2526 (1997)]

direct observation of fractional charge ?!

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?

0

lim ( ), (0)i tS dte I t I

2 2 **

*

24 2 (1 ) coth

2B

BB

k Te e e US k T e U t t

h h k T e U

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

state devices

Reminder of TLL model

22

2

1 ( )( )

2Fv x

H dg

x xx

Low energy fixed point Hamiltonian:

0 1g interaction parameter:

Electron field operator (in bosonization):

2 / ( ) ( )1( )

2F pip k x N x L x i x

p px U ea

Klein factors

( )2

hx

x

Impurity in a TLL

20

0

( ) ( ) cos 2 (0)Fk FI

x

W kWH dxW x x

x

can be scaled away bya unitary transformation

dominant contributionat low energies

Fixed point Hamiltonian:2

22

1 ( )( ) cos 2 (0)

2Fv x

H dx xg x

• corresponds to tunneling of quasiparticles with charge e*=eg• bears a resemblance to the boundary sine-Gordon Hamiltonian

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))

2

0

eG g

h

2

0

eG

h

(0)U

eUH

[derived by: Maslov and Stone, Ponomarenko, Safi and Schulz, Kawabata, Shimizu, etc., using different methods and ways of thinking about the problem]

Shot noise

0lim ( ), (0)i tS dte I t I

Perturbative calculations in Keldysh formalism give:

2S e I

02S eg I I

strong backscattering limit

weak backscattering limit

*e eg

[Kane and Fisher, PRL 72, 724 (1994)]

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)]

Exact solution for g=1/2Expression for the shot noise at finite temperature:

1/(1 )Be

2 2 4(1 ) (1 ) | | ( ) | |S

d e f f f f Q f f QT

with the effective transmission coefficient

22( )

1| |

1 BQ

e

The right(+) and left(-) moving quasiparticles obey the distribution function

(exp( ) ) / 2

1

1 U Tf

e

[Fendley and Saleur, PRB 54, 10845 (1996)]

Heuristic formulas for the noise

Simple IPM:

Advanced IPM:

constant transmission

2 **

*

24 2 (1 ( )) coth

2B

BB

k Te e US k T e I t U

h k T e U

* 2( )

e h dIt U

e e dU

2 2 **

*

24 2 (1 ) coth

2B

BB

k Te e e US k T e U t t

h h k T e U

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).]

g

Comparison of heuristic formulas and exact solution for the case g=1/2

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strong backscattering limit(t=0.14)

weak backscattering limit(t=0.95)

[Glattli, Roche, Saleur, and Trauzettel, in preparation]

2. Part :Shot noise of non-chiral TLL

systems(i.e. carbon nanotubes, cleaved edge

overgrowth quantum wires, etc.)

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

2

22

1 ( )( ) cos 4 (0)

2 ( )F

U

v xH dx

xgH

xx

( ) ( )U x

eH dxU x x

shifts band bottom in leads electroneutrality

[Maslov and Stone; Ponomarenko; Safi and Schulz, PRB (1995); Furusaki and Nagaosa, PRB (1996)]

Inhomogeneous correlation functionequations of motion: 2

2

1( , ) 0

( )t x x x tg x

( , ) ( ) ( )x t x t find the eigenfunctions of the inhomogeneous Laplacian

*, ,

1,2

( ) ( )( , , ) ( , ) ( ,0)

4s s i t

s

x yiG x y t x t y d e

Special situation x=y:

2 2 2 2| | | |

2 2 2 2

( , , ) ( , ,0)

( / ) ( ) ( 2 / )ln ln

4 ( 2 / )even odd

m m

m m

iG x x t iG x x

g it L m i m x L

m m x L

UV cutoff

Calculation of the current

( , )t

eI x t

Current (in bosonization):

( , ) ( , ) ( , )px t x t x t 2

2( )| |,| |

22( )( , )

2| |,| |

22

F

p

F

U V Lx x

ve U V tx t

eg V Lx x

v

2

( )e

I U Vh

obtain the four-terminal voltage drop V(U) by requiring that ( , ) 0t x t

[see e.g., Egger and Grabert, PRL 77, 538 (1996); 80, 2255(E)]]

particular solution of the motion determined by the fullaction (based on radiative boundary condition approach)

Results for the backscattered current2

0BS

eI I I V

h

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1/ 4g / Fu eUgL v

st oscBS BS BSI I I

2 1

( , ) 2 2

g

st BBS

B

e eUI

C g g

/(1 )g g

B a

order 2calculation

[Dolcini, Grabert, Safi, and Trauzettel, in preparation]

Calculation of the true shot noise

2

0lim ( , ), ( ,0)i t

t t

eS dte x t x

path integral with respect to the full action

evaluation of the path integral at order 2 yields2 2

2 BSeS I

• 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)]

2 2

)( () /2 BSLS eI

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 :

2

2 2

2

1 (1 )cos( / )( / )L

L

g

g g

at high frequencies and/or for long quantum wires, it should indeed be possible to observe a fractional charge

2 2

2 BSS egI

2 2

2 BSS eI

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)

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]

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)