1. 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 … ) (QHE Graphene)
Valentin Rodriguez ( Th ) (FQHE)
H. Perrin ( Post-Doc. 99 ) (FQHE)
Laurent Saminadayar ( Th ) (FQHE)
(+ L.-H. Bize, J. Ségala, E. Zakka-Bajani, 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 (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)
ENS-Paris

2. OUTLINE Fractional Quantum Hall effect Edges as Chiral LL:
Carbone Nanotubes signatures of T-LL:
(on going or foreseen experimental projects)

3. Integer Quantum Hall Effect
Edwin Hall
1879 V
Hall I
V xx
1980
K. von Klitzing
G. Dorda
M. Pepper
B (Tesla) E
Landau levels

4. Fractional Quantum Hall Effect
(1982)
(D.C. Tsui, H. Störmer, and A.C. Gossard, 1982)
Laughlin’s predictions:
for filling factors n:
(1996) D m
1/3
2/3 1
FQHE Gap : fundamental incompressibility due to interactions
(different from IQHE incompressibility due to Fermi statistics)
( FQHE )
( IQHE )

5. i.e. 3 flux quanta (or 3 states) for 1 electron
Laughlin quasiparticles
quasi-hole wavefunction at z = z a
Example : n = 1/3
i.e. 3 flux quanta (or 3 states) for 1 electron
a quasi-hole excitation = to add a quantum flux
= to create a charge (- e / 3)
Gap D
- e / 3
single particle
wavefunction :
Laughlin trial wavefunction for n = 1/3, 1/5, … :
(Ground State)
- satisfies Fermi statistics
- minimizes interactions
- uniform incompressible quantum liquid
fractionally charged quasiparticles obey fractional statistics
anyons !!!

6. current flows only on the edges (edge channels)
confining
potential
(Landau levels)
electron drift velocity
edge
channels

7. OUTLINE Fractional Quantum Hall effect Edges as Chiral LL:
probing quasiparticles via tunneling experiments
Carbone Nanotubes signatures of T-LL:
on going or foreseen experimental projects

8. Laughlin quasiparticles
probing quasiparticles via tunneling experiments,
two different approaches:
1) non-equilibrium tunneling current measurements:
probes excitations above the ground state
tunneling density of states :
how quasiparticles are created
2) shot-noise associated with the
tunneling current:
probes excitations above the ground states :
direct measure of quasi-particle charge
B (Tesla)
e/3
metal e
=1/3
e/3
e/3 q
q = e/3
e/3
=1/3
-e/3 e
Laughlin quasiparticles
on the edge

9. Tunneling electrons into Tomonaga-Luttinger liquids
Haldane (1979)
1-D fermions short range interactions
(connection with exactly integrable quantum models: Calogero, Sutherland, …)
Tunneling electrons into Tomonaga-Luttinger liquids
tunelling density of states depends on energy
 differential conductance is non-linear with voltage
non-linear conductance:
(métal) e
plasmon
(1D conductor)
plasmon
example : SW Carbone Nanotube

10. 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)
+ field quantization: e
+ electron creation operator on the edge
+ Fermi statistics :

11. 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)
+ field quantization: e
+ electron creation operator on the edge
+ Fermi statistics :
properties of a Luttinger liquid with g = n

12. of the current / voltage
tunneling from a metal to a FQHE edge
power law variation
of the current / voltage
Chiral-Luttinger prediction:
n+ GaAs
2 DEG V e
A.M. Chang (1996)
also observed :
(voltage and temperature play the same role)

13. 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)

14. (Quantum Point Contact)
tunneling between FQHE edges
2D electrons
GaAs
GaAlAs Si +
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)

15. tunneling between FQHE edges
high barrier
(doubled)
energy
energy
weak barrier
low energy
large energy
even the weakest barrier
leads to strong reflection
at low energy !

16. tunneling between FQHE edges (TBA solution of the B.Sine-Gordon model)
folded into:
kink / anti-kink (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

17. tunneling between FQHE edges
(impurity, strength TB )
eV << TB
eV >> TB
Numerical calculation of G(V) using the exact solution by FLS (1996)
(P.Roche + C. Glattli 2002 )

18. tunneling between FQHE edges : experimental comparison
energy
very weak barrier

19. renormalization fixed point limit
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

20. Finite temperature calculation using the TBA
solution of the boundary Sine-Gordon model
(Saclay 2000)
(scaling law experimentally observed (Saclay 1998) )
to observe exponent =4 one needs very low temperature and conductance 10-4 X e2/3h !
weak barrier

21. 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 10-4 X e2/3h !
weak barrier

22. Laughlin quasiparticles
probing quasiparticles via tunneling experiments,
two different approaches:
1) non-equilibrium tunneling current measurements:
probes excitations above the ground state
tunneling density of states :
how quasiparticles are created
2) shot-noise associated with the
tunneling current:
probes excitations above the ground states :
direct measure of quasi-particle 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
q = e/3
Laughlin quasiparticles
on the edge

23. 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
2 limiting cases:

24. (ballistic conductor)
quantum point contact (B=0)
Gate
2-D
electron
gas
Gate
(ballistic conductor)
(Saclay 1996)
0,0
0,2
0,4
0,6
0,8
1,0 ( ) 1 - T 2 +
Fano reduction factor
Conductance 2e² / h .8 .6 .4 .2
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)

25. Shot Noise in IQHE regime
strong barrier : e V
 = 1
 = 1
transmitted (D) reflected (1-D) e e
(rarely transmitted electrons)
(incoming electrons)
weak barrier :
(rarely transmitted holes) e

26. Shot Noise in IQHE regime
strong barrier :
transmitted (D) reflected (1-D) e V
 = 1/3
 = 1/3 e e e e
(rarely transmitted electrons)
(incoming electrons)
e/3
e/3
weak barrier :
(rarely transmitted holes)
e/3

27. Direct evidence of fractional charge
L. Saminadayar et al. PRL (1997).
De Picciotto et al. Nature (1997)
e / 3
n = 1/3
charge q=e/3
n = 2
charge q=e
measure of the anti-correlated transmitted X reflected
current fluctuations (electronic Hanbury-Brown Twiss)

28. From fractional to integer charges
charge e/3
charge e ??
V. Rodriguez et al (2000)

29. From fractional to integer charges

30. extremely good ! numerical calculation of the finite temperature
shot noise
P. Fendley and H. Saleur,
Phys. Rev. B 54, (1996)
exact
solution
(Bethe
Ansatz)
dotted line:
empirical binomial noise
formula for backscattered
e/3 quasiparticles
(P.Roche + C. Glattli 2002 )
extremely good !

31. 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, (2004)

32. OUTLINE Fractional Quantum Hall effect Edges as Chiral LL:
probing quasiparticles via tunneling experiments
Carbone Nanotubes signatures of T-LL:
on going or foreseen experimental projects

33. graphene energy band structure

34. 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
Non-linear conductance:
plasmon
plasmon
provided kT or eV < hvF / L

35. Luttinger Liquid effects in Single Wall Nanotubes

36. Luttinger Liquid effects in Single Wall Nanotubes
Observation of LL effects requires

37. Luttinger-Liquid behavior in Crossed Metallic Single-Wall Nanotubes

38. Luttinger-Liquid behavior in Crossed Metallic Single-Wall Nanotubes
B. Gao, A. Komnik, R. Egger, D.C. Glattli and A. Bachtold, Phys. Rev. Lett. 92, (2004)
Mesoscopic Physics group, Lab. P. Aigrain, ENS Paris
~700 nm
1D conductor :
quantum transport + e-e interaction
lead to non-linear I-V for tunneling
from one nanotube to the other (zero-bias anomaly):
differential tube-tube conductance e V I
g = 0.16

39. A current flowing through NT ‘ B ’ changes in a
OBSERVED PREDICTED
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, (2004)

40. OUTLINE Fractional Quantum Hall effect Edges as Chiral LL:
probing quasiparticles via tunneling experiments
Carbone Nanotubes signatures of T-LL:
on going or foreseen experimental projects

41. Possible future experimental investigations
High frequency shot noise of fractional charges (FQHE in GaAs/GaAlAs)
see arXiv: by C. Bena and I. Safi
shot noise singularity at e*V/h
Carbone Nanotubes
shot noise : fractional charges observation would requires >THz
measurements
FQHE in Graphene ?
E. Zakka-Bajani PRL 2007
R12,42
electrons
holes
K. Bennaceur (Saclay SPEC)