ENS-Paris
This presentation is the property of its rightful owner.
Sponsored Links
1 / 41

ENS-Paris PowerPoint PPT Presentation


  • 123 Views
  • Uploaded on
  • Presentation posted in: General

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 )

Download Presentation

ENS-Paris

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Ens paris

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)


Ens paris

OUTLINE

  • Fractional Quantum Hall effectEdges as Chiral LL:

  • Carbone Nanotubes signatures of T-LL:

  • (on going or foreseen experimental projects)


Ens paris

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)


Ens paris

(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


Ens paris

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


Ens paris

electron drift velocity

edge

channels

current flows only on the edges (edge channels)

confining

potential

(Landau levels)


Ens paris

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


  • Ens paris

    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


    Ens paris

    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


    Ens paris

    + 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)


    Ens paris

    (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


    Ens paris

    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)


    Ens paris

    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)


    Ens paris

    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)


    Ens paris

    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


    Ens paris

    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


    Ens paris

    (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 )


    Ens paris

    tunneling between FQHE edges : experimental comparison

    energy

    0

    very weak barrier


    Ens paris

    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


    Ens paris

    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


    Ens paris

    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


    Ens paris

    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


    Ens paris

    ( 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


    Ens paris

    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..


    Ens paris

    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


    Ens paris

    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


    Ens paris

    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)


    Ens paris

    From fractional to integer charges

    chargee/3

    charge e

    ??

    V. Rodriguez et al (2000)


    Ens paris

    From fractional to integer charges


    Ens paris

    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 !


    Ens paris

    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)


    Ens paris

    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


  • Ens paris

    graphene energy band structure


    Ens paris

    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


    Ens paris

    Luttinger Liquid effects in Single Wall Nanotubes


    Ens paris

    Luttinger Liquid effects in Single Wall Nanotubes

    Observation of LL effects requires


    Ens paris

    Luttinger-Liquid behavior in Crossed Metallic Single-Wall Nanotubes


    Ens paris

    ~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


    Ens paris

    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)


    Ens paris

    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


  • Ens paris

    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)


  • Login