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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]
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Why is it interesting?
[de-Picciotto et al., Nature 389, 162 (1997)][Saminadayaret al., PRL 79, 2526 (1997)]
direct observation of fractional charge ?!
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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
!
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1. Part :
Interpretation of shot noiseexperiments on FQH edge
state devices
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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
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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
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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]
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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)]
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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)]
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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)]
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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 !
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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]
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2. Part :
Shot noise of non-chiral TLLsystems(i.e. carbon nanotubes, cleaved edge
overgrowth quantum wires, etc.)
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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)]
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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
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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)
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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]
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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)]
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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
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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)
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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]
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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)