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)
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