present trauzettel

8/6/2019 Present Trauzettel

1/22

Current noise in 1D electron systemsISSP International Summer School

August 2003

Bjrn Trauzettel

Albert-Ludwigs-Universitt Freiburg, Germany

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


2/22

Why is it interesting?

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

direct observation of fractional charge ?!


3/22

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 Saminadayaret al. in terms of

the Tomonaga-Luttinger-Liquid (TLL) model?

a_0

lim ( ), (0)i t

S dte I t I [

[ p! ( (

2 2 **

*

24 2 (1 ) co th

2

B

B

B

k Te e e U S k T e U t t

h h k T e UR R

!


4/22

1. Part :

Interpretation of shot noiseexperiments on FQH edge

state devices


5/22

Reminder of TLL model

2

2

2

1 ( )( )

2

Fv xH dg

x xx

. x ! 4 x

Low energy fixed point Hamiltonian:

0 1g interaction parameter:

Electron field operator (in bosonization):

2 / ( ) ( )1( )

2

F pip k x N x L x i x

p px U ea

T T . T J ]

T

!

Klein factors

( )2

hx

x

J

T

x4 !

x


6/22

Impurity in a TLL20

0

( ) ( ) cos 2 (0)Fk F

I

x

W kWdxW x x

x

. V T .

TT !

x ! ! x

can be scaled away by

a unitary transformationdominant contribution

at low energies

Fixed point Hamiltonian:

2

2

21 ( )( ) cos 2 (0)

2

Fv xH dx xg x

. P T. x ! 4 x

corresponds to tunneling of quasiparticles with charge e*=eg

bears a resemblance to theboundary sine-Gordon Hamiltonian


7/22

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 eU

H .T!

[derived by: Maslov and Stone, Ponomarenko, Safi and Schulz, Kawabata,

Shimizu, etc., using different methods and ways of thinking about the problem]


8/22

Shot noisea_

0lim ( ), (0)

i tS dte I t I [

[ p! ( (

Perturbative calculations in Keldysh formalism give:

2S e I!

02S eg I I !

strongbackscattering limit

weakbackscattering limit

*e eg!

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


9/22

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 statisticsthat depend on the energy and the interactions ( TBAequations)

local operators act in a quite complicated fashion on thequasi particle basis

however, the total charge operator acts diagonally on this

basis calculation of the current and the noise is not somessy

apply the Landauer-Bttiker formalism to these particles

[Fendley, Ludwig, and Saleur, PRL +PRB (1995-96)]


10/22

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

1/(1 )BeU RP w

? A_ a2 2 4(1 ) (1 ) | | ( ) | |S d e f f f f Q f f QT

UU !

with the effective transmission coefficient

2

2( )

1| |

1 BQ

e U U !

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

(exp( ) ) / 2

1

1U T

fe

Us!

m

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


11/22

Heuristic formulas for the noiseSimple IPM:

Advanced IPM:

constant transmission

2 **

*

24 2 (1 ( )) coth

2

B

B

B

ke e US k e I tU

h k e UR

!

* 2

( )e h dI

tUe e dU

!

2 2 **

*

24 2 (1 ) coth

2

B

B

B

ke e e US k e U t t

h h k e UR R

!

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).]

gR !


12/22

Comparison of heuristic formulas and exact solution

for the case g=1/2

0 2 4 6

V/T

1.75

2.25

2.75

S

/T

exact solution with B

= 2.0

advanced IPM with e*=e/2

advanced IPM with e*=e

simple IPM with e*=e/2

simple IPM with e*=e

0 2 4 6

V/T

0

0.5

1

1.5

S/T

exact solution with B=2.0

advanced IPM with e*=e/2

advanced IPM with e*=e

simple IPM with e*=e/2

simple IPM with e*=e

strong backscattering limit

(t=0.14)

weak backscattering limit

(t=0.95)

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


13/22

2. Part :

Shot noise of non-chiral TLLsystems(i.e. carbon nanotubes, cleaved edge

overgrowth quantum wires, etc.)


14/22

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

2

2

1 ( )( ) cos 4 (0)

2 ( )

FU

v xH dx

xgH

xx

.P T.

x ! 4 x

( ) ( )U x

eH dxU x x.

T! x

shifts band bottom in

leads electroneutrality

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


15/22

Inhomogeneous correlation functionequations of motion: 2

2

1( , ) 0

( )t x x

x tg x

.

x x x !

( , ) ( ) ( )x t x t. U J! find the eigenfunctions of theinhomogeneous Laplacian

*

, ,

1,2

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

4

s s i t

s

x yi x y t x t y d e

[ [ [U U

. . [T [

"

!

;! !

Special situation x=y:

2 2 2 2

2 2 2 2

( , , ) ( , ,0)

( / ) ( ) ( 2 / )ln ln

4 ( 2 / )even odd

m m

m m

i x x t i x x

g it L m i m x L

m m x L

E E XK K

T E E

" "

!

UV cutoff


16/22

Calculation of the current( , )t

eI x t.

T! xCurrent (in bosonization):

( , ) ( , ) ( , )px t x t x t. U J! 2

2( )| |, | |

22( )( , )2

| |, | |22

F

p

F

U V Lx x

ve U V t x teg V L

x xv

TUT

T

" !

2

( )e

I U V h

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

t x tJx !

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

particular solution of the motion determined by the full

action (based on radiative boundary condition approach)


17/22

Results for the backscattered current20B S

e I I I V

h! !

0 5 10 15 20 25 30

0.0

0.5

1.0

1.5

2.0

IBS

u

1/4g! /F

u eUgL v!

st osc

BS BS BS I I I }

2 1

( , ) 2 2

g

st BBS

B

e eUI

C g g

P T

E P

! +

/(1 )g g

BaP P P !

orderP2

calculation

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


18/22

Calculation of the true shot noise_ a

2

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

t t

eS dte x t x[

[J J

T p! x x

path integral with

respect to the full action

evaluation of the path integral at orderP2 yields2 2

2B SeS I

P P!

no visibility of fractional charge in the weakbackscattering 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)]


19/22

2 2

)( () /2BSLS eI

P P[ [[ +!

What happens at higher frequencies?

We still talk about shot noise at zero temperature, but welook at two regimes:

L


20/22

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)


21/22

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]


22/22

In collaboration with:

Christian Glattli (CEA Saclay, France)

Patrice Roche (CEA Saclay, France)

Hubert Saleur(CEA Saclay, France)

Fabrizio Dolcini (Freiburg, Germany)

Reinhold Egger(Dsseldorf, Germany)

Hermann Grabert (Freiburg, Germany)Ins Safi (Orsay, France)

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