ENSParis. Experiments on Luttinger liquid properties of Fractional Quantum Hall effect and Carbon Nanotubes. Christian Glattli CEA Saclay / ENS Paris). Nanoelectronic Group (SPEC, CEA Saclay) Patrice Roche ( join in 2000 ) (FQHE) Fabien Portier ( join in 2004 )
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Experiments on Luttinger liquid properties of
Fractional Quantum Hall effect
and Carbon Nanotubes.
Christian Glattli CEA Saclay /ENS Paris)
Nanoelectronic Group (SPEC, CEA Saclay)
Patrice Roche ( join in 2000) (FQHE)
Fabien Portier (join in 2004)
Keyan Bennaceur (Th. 07  … ) (QHEGraphene)
Valentin Rodriguez ( Th. 97  00 ) (FQHE)
H. Perrin ( PostDoc. 99 ) (FQHE)
Laurent Saminadayar ( Th. 94  97) (FQHE)
(+ L.H. Bize, J. Ségala, E. ZakkaBajani, P. Roulleau, …)
Mesoscopic Physics Group (LPA, ENS Paris)
J.M. Berroir
B. Plaçais
A. Bachtold (now in Barcelona) (LL in CNT)
T. Kontos (Shot noise in CNT)
Gao Bo (PhD 2003  2006 (LL in CNT)
L. Herrmann (Diploma arbeit 07)
+ Th. Delattre ( Shot noise in CNT)
( +G. Gève, A. Mahé, J. Chaste, C. Feuillet Palma, B. Bourlon)
V xx
V
Hall
I
I
I
Edwin Hall
1879
K. von Klitzing
G. Dorda
M. Pepper
1980
E
Landau levels
Integer Quantum Hall Effect
B (Tesla)
(1982)
Laughlin’s predictions:
for filling factors n:
(D.C. Tsui, H. Störmer, and A.C. Gossard, 1982)
(1996)
( IQHE )
m
( FQHE )
D
0
1/3
2/3
1
FQHE Gap : fundamental incompressibility due to interactions
(different from IQHE incompressibility due to Fermi statistics)
Fractional Quantum Hall Effect
Example :n=1/3
i.e. 3 flux quanta (or 3 states) for 1 electron
single particle
wavefunction :
Gap D
Laughlin trial wavefunction for n = 1/3, 1/5, … :
(Ground State)
 e / 3
 satisfies Fermi statistics
 minimizes interactions
 uniform incompressible quantum liquid
a quasihole excitation = to add a quantum flux
= to create a charge ( e / 3)
quasihole wavefunction at z = z a
fractionally charged quasiparticles obey fractional statistics
anyons !!!
Laughlin quasiparticles
edge
channels
current flows only on the edges (edge channels)
confining
potential
(Landau levels)
e/3
=1/3
e/3
=1/3
e/3
q = e/3
Laughlin quasiparticles
on the edge
q
probing quasiparticles via tunneling experiments,
two different approaches:
1) nonequilibrium tunneling current measurements:
probes excitations above the ground state
tunneling density of states :
how quasiparticles are created
2) shotnoise associated with the
tunneling current:
probes excitations above the ground states :
direct measure of quasiparticle charge
B (Tesla)
e/3
metal
e
=1/3
e/3
e/3
Tomonaga (1950), Luttinger (1960)
Haldane (1979)
1D fermions short range interactions
(connection with exactly integrable quantum models: Calogero, Sutherland, …)
Tunneling electrons into TomonagaLuttinger liquids
tunelling density of states depends on energy
differential conductance is nonlinear with voltage
nonlinear conductance:
(métal)
e
plasmon
(1D conductor)
plasmon
example : SW Carbone Nanotube
+ electron creation operator on the edge
e
+ Fermi statistics :
Tunneling into Chiral Luttinger liquid (FQHE regime)
X.G. Wen (1990)
periphery deformation of 1/3 incompressible
FQHE electron liquid
Classical hydrodynamics
(excess charge density / length)
(excess charge density / length)
+ field quantization:
+ electron creation operator on the edge
e
+ Fermi statistics :
properties of a Luttinger liquid with g = n
Tunneling into Chiral Luttinger liquid (FQHE regime)
X.G. Wen (1990)
periphery deformation of 1/3 incompressible
FQHE electron liquid
Classical hydrodynamics
2 DEG
n+ GaAs
V
tunneling from a metal to a FQHE edge
power law variation
of the current / voltage
ChiralLuttinger prediction:
A.M. Chang (1996)
also observed :
(voltage and temperature play the same role)
tunneling from a metal to a FQHE edge
Simplest theory predicts for
power laws are stille observed as expected
but exponent found is different.
Not included
interaction of bosonic mode dynamics with finite
conductivity in the bulk
 long range interaction
 acuurate description of the edge in real sample.
Grayson et al. (1998)
+
GaAlAs
GaAs
tunneling between FQHE edges
2D electrons
Atomically controlled epitaxial growth
GaAs/Ga(Al)As heterojunction
CLEAN 2D electron gas
heterojunction
100 nm
constriction
(Quantum Point Contact)
200nm
(top view )
(edge channel)
even the weakest barrier
leads to strong reflection
at low energy !
weak barrier
large energy
energy
tunneling between FQHE edges
high barrier
(doubled)
energy
tunneling between FQHE edges (TBA solution of the B.SineGordon model)
folded into:
kink / antikink (charged solitons ) in the phase field f(x,t)
breather (neutral soliton )
thermodynamic Bethe Ansatz self consistent equations
Expression of the current
P. Fendley, A. W. W. Ludwig, and H. Saleur,
Phys. Rev. Lett. 74, 3005 (1995); 75, 2196 (1995);
… similar calculation for shot noise
eV >> TB
eV << TB
Numerical calculation of G(V) using the exact solution by FLS (1996)
tunneling between FQHE edges
(P.Roche + C. Glattli 2002 )
tunneling between FQHE edges : experimental comparison
scaling V/T is OK
… but dI/dV varies as the second
instead of the fourth power of V( or T)
predicted by perturbative renormalization
approach.
solid line:
renormalization fixed point limit
Finite temperature calculation using the TBA
solution of the boundary SineGordon model
(Saclay 2000)
(scaling law experimentally observed (Saclay 1998) )
to observe exponent =4 one needs very low temperature and conductance 104 X e2/3h !
weak barrier
Finite temperature calculation using the
Fendley, Ludwig, Saleur (1995) exact solution
e/3
e
(Saclay 2000)
(scaling law experimentally observed (Saclay 1998) )
to observe exponent =4 one needs very low temperature and conductance 104 X e2/3h !
weak barrier
probing quasiparticles via tunneling experiments,
two different approaches:
1) nonequilibrium tunneling current measurements:
probes excitations above the ground state
tunneling density of states :
how quasiparticles are created
2) shotnoise associated with the
tunneling current:
probes excitations above the ground states :
direct measure of quasiparticle charge
B (Tesla)
e/3
metal
e
=1/3
e/3
e/3
e
e/3
=1/3
e/3
=1/3
e/3
q = e/3
Laughlin quasiparticles
on the edge
( t )
( r )
2 limiting cases:
The binomial statistics of Shot Noise (no interactions)
incoming current :
(noiseless thanks
to Fermi statistics)
transmitted current :
current noise in B.W. Df :
Variance of partioning
binomial statistics
1
.8
.6
.4
.2
0
(
)

1
T
0,8
1
0,6
(
)
Fano reduction factor

T
1
T
2
2
0,4
+
1
T
2
0,2
0,0
0. 0.5 1. 1.5 2. 2.5
Conductance 2e² / h
quantum point contact (B=0)
Gate
2D
electron
gas
Gate
(ballistic conductor)
(Saclay 1996)
first mode :
slope ~ (1  D1 )
Kumar et al. PRL (1996)
M. I. Reznikov et al., Phys. Rev. Lett. 75 (1995) 3340.
A. Kumar et al. Phys. Rev. Lett. 76 (1996) 2778..
Shot Noise in IQHE regime
strong barrier :
e
= 1
= 1
transmitted (D) reflected (1D)
e
e
(rarely transmitted electrons)
(incoming electrons)
weak barrier :
(rarely transmitted holes)
e
e
Shot Noise in IQHE regime
strong barrier :
transmitted (D) reflected (1D)
e
= 1/3
= 1/3
e
e
e
(rarely transmitted electrons)
e/3
e/3
(incoming electrons)
weak barrier :
(rarely transmitted holes)
e/3
Direct evidence of fractional charge
L. Saminadayar et al. PRL (1997).
De Picciotto et al. Nature (1997)
n = 1/3
charge q=e/3
n = 2
charge q=e
measure of the anticorrelated transmitted X reflected
current fluctuations (electronic HanburyBrown Twiss)
the finite temperature
shot noise
P. Fendley and H. Saleur,
Phys. Rev. B 54, 10845 (1996)
exact
solution
(Bethe
Ansatz)
dotted line:
empirical binomial noise
formula for backscattered
e/3 quasiparticles
(P.Roche + C. Glattli 2002 )
extremely good !
heuristic formula for shot noise
(binomial stat. noise
of backscattered qp )
(binomial stat. noise
of transmitted electrons )
e* as free parameter
B. Trauzettel, P. Roche, D.C. Glattli, H. Saleur
Phys. Rev. B 70, 233301 (2004)
Luttinger Liquid effects in Single Wall Nanotubes
Electron tunneling into a SWNT excites
1D plasmons in the nanotubes
giving rise to Luttinger liquid effects
SWNT
e
Nonlinear conductance:
plasmon
plasmon
provided kT or eV < hvF / L
Luttinger Liquid effects in Single Wall Nanotubes
Observation of LL effects requires
~700 nm Nanotubes
e
V
I
differential tubetube conductance
LuttingerLiquid behavior in Crossed Metallic SingleWall Nanotubes
B. Gao, A. Komnik, R. Egger, D.C. Glattli and A. Bachtold, Phys. Rev. Lett. 92, 216804 (2004)
Mesoscopic Physics group, Lab. P. Aigrain, ENS Paris
1D conductor :
quantum transport + ee interaction
lead to nonlinear IV for tunneling
from one nanotube to the other (zerobias anomaly):
g = 0.16
OBSERVED NanotubesPREDICTED
A current flowing through NT ‘ B ’ changes in a
non trivial way the conductance of NT ‘ A ’
additonal demonstration that Luttinger theory is
the good description of transport in CNT at large V
B. Gao, A. Komnik, R. Egger, D.C. Glattli and A. Bachtold,
Phys. Rev. Lett. 92, 216804 (2004)
OUTLINE Nanotubes
E. ZakkaBajani PRL 2007 Nanotubes
Possible future experimental investigations
R12,42
electrons
holes
K. Bennaceur (Saclay SPEC)