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ENS-Paris. 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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ENS-Paris

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 ( Post-Doc. 99 ) (FQHE)

Laurent Saminadayar ( Th. 94 - 97) (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 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)


OUTLINE

  • Fractional Quantum Hall effectEdges as Chiral LL:

  • Carbone Nanotubes signatures of T-LL:

  • (on going or foreseen experimental projects)


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 quasi-hole excitation = to add a quantum flux

= to create a charge (- e / 3)

quasi-hole wavefunction at z = z a

fractionally charged quasiparticles obey fractional statistics

anyons !!!

Laughlin quasiparticles


electron drift velocity

edge

channels

current flows only on the edges (edge channels)

confining

potential

(Landau levels)


OUTLINE

  • Fractional Quantum Hall effectEdges as Chiral LL:

    • probing quasiparticles via tunneling experiments

  • Carbone Nanotubes signatures of T-LL:

  • on going or foreseen experimental projects


  • e

    -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) 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


    Tomonaga (1950), Luttinger (1960)

    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


    + field quantization:

    + 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


    e

    2 DEG

    n+ GaAs

    V

    tunneling from a metal to a FQHE edge

    power law variation

    of the current / voltage

    Chiral-Luttinger 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)


    Si

    +

    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)


    low energy

    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.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


    (impurity, strength TB )

    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

    energy

    0

    very weak barrier


    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 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


    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


    q

    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 = e/3

    Laughlin quasiparticles

    on the edge


    ( i )

    ( 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


    1,0

    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

    2-D

    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..


    V

    Shot Noise in IQHE regime

    strong barrier :

    e

    = 1

    = 1

    transmitted (D)reflected (1-D)

    e

    e

    (rarely transmitted electrons)

    (incoming electrons)

    weak barrier :

    (rarely transmitted holes)

    e


    V

    e

    Shot Noise in IQHE regime

    strong barrier :

    transmitted (D)reflected (1-D)

    e

    = 1/3

    = 1/3

    e

    e

    e

    (rarely transmitted electrons)

    e/3

    e/3

    (incoming electrons)

    weak barrier :

    (rarely transmitted holes)

    e/3


    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 anti-correlated transmitted X reflected

    current fluctuations (electronic Hanbury-Brown Twiss)


    From fractional to integer charges

    chargee/3

    charge e

    ??

    V. Rodriguez et al (2000)


    From fractional to integer charges


    numerical calculation of

    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)


    OUTLINE

    • Fractional Quantum Hall effectEdges as Chiral LL:

      • probing quasiparticles via tunneling experiments

  • Carbone Nanotubes signatures of T-LL:

  • on going or foreseen experimental projects


  • graphene energy band structure


    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


    Luttinger Liquid effects in Single Wall Nanotubes


    Luttinger Liquid effects in Single Wall Nanotubes

    Observation of LL effects requires


    Luttinger-Liquid behavior in Crossed Metallic Single-Wall Nanotubes


    ~700 nm

    e

    V

    I

    differential tube-tube conductance

    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, 216804 (2004)

    Mesoscopic Physics group, Lab. P. Aigrain, ENS Paris

    1D conductor :

    quantum transport + e-e interaction

    lead to non-linear I-V for tunneling

    from one nanotube to the other (zero-bias anomaly):

    g = 0.16


    OBSERVEDPREDICTED

    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

    • Fractional Quantum Hall effectEdges as Chiral LL:

      • probing quasiparticles via tunneling experiments

  • Carbone Nanotubes signatures of T-LL:

  • on going or foreseen experimental projects


  • E. Zakka-Bajani PRL 2007

    Possible future experimental investigations

    • High frequency shot noise of fractional charges (FQHE in GaAs/GaAlAs)

    • seearXiv:0705.0156 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 ?

    R12,42

    electrons

    holes

    K. Bennaceur (Saclay SPEC)


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