ens-paris

ENS-Paris

Experiments on Luttinger liquid properties of Fractional Quantum Hall effect

and Carbon Nanotubes.Christian Glattli CEA Saclay / ENS Paris)

- e

e/3 e/3

e/3e

B > Tesla

Nanoelectronic Group (SPEC, CEA Saclay)

Patrice Roche ( join in 2000) (FQHE)Fabien Portier (join in 2004)Keyan Bennaceur (Th. 07 - … ) (QHE Graphene)

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. BerroirB. PlaçaisA. 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 effect Edges as Chiral LL:

• Carbone Nanotubes signatures of T-LL:

• (on going or foreseen experimental projects)

Integer Quantum Hall Effect

1980

K. von KlitzingG. DordaM. PepperIntegere

hR Hall

12

e

hn 0 quantumflux ofdensity

ne

hB

sHall n

n

e

hR

2

nIntegern s )(

zBvE Hall ˆ

sHall ne

BR

Edwin Hall 1879

VHall

II

V xx

I

E

degeneraten

C

Landau levels

0

B (Tesla)

1

2

e

hR Hall

Laughlin’s predictions:

fractionsr denominato odd ... 3/7, 2/7, 2/5, 1/5, 2/3, 1/3,n

ns

for filling factors

(1996)

0 1/3 2/3 1

FQHE Gap : fundamental incompressibility due to interactions

(different from IQHE incompressibility due to Fermi statistics)

( FQHE )

( IQHE )

Fractional Quantum Hall Effect

(1982)

(D.C. Tsui, H. Störmer, and A.C. Gossard, 1982)

ii

k

Njiij

k

zzz212.SG.

12

1 4

1exp

2

4

1exp

!22

1zz

m

m

mm

single particle wavefunction :

Laughlin trial wavefunction for = 1/3, 1/5, … :(Ground State)

- satisfies Fermi statistics- minimizes interactions- uniform incompressible quantum liquid

quasi-hole wavefunction at z = z a

ii

k

Njiij

Niaja

ol

k

zzzzzz212.eh

12

1 4

1exp

Example : i.e. 3 flux quanta (or 3 states) for 1 electron

0

a quasi-hole excitation = to add a quantum flux

= to create a charge (- e / 3)

0

Gap

Cl

e

2

- e / 3

fractionally charged quasiparticles obey fractional statistics

az

bz

),(3

exp),( 22ab

holesba

holes zzizz

anyons !!!

Laughlin quasiparticles

confining potential

C2

1

C2

3

C2

5

(Landau levels)

)(YU

)()2/1( YUnH Cn

Y

YU

eBvdrift

)(1

electron drift velocity2

edgechannels

h

eG

2

current flows only on the edges (edge channels)

OUTLINE


• probing quasiparticles via tunneling experiments


• on going or foreseen experimental projects

metal =1/3 e

e/3e/3

) liquid Luttinger:later also (see

char. V-Ilinear -non)( 3VVI

e/3

IffSI I 2)0(2 q q = e/3

e/3=1/3 =1/3

-e/3

e/3

e

Laughlin quasiparticleson the edge

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)

probing quasiparticles via tunneling experiments,two different approaches:

Tomonaga (1950), Luttinger (1960)Haldane (1979)

1-D fermions short range interactions

(connection with exactly integrable quantum models: Calogero, Sutherland, …)

e

plasmon

non-linear conductance:

)2( 1

ggVdV

dI

22g

v

vg

F

F

plasmon

example : SW Carbone Nanotube

(métal)

(1D conductor)

dVdI /

T.D.O.S.

V voltage

tunelling density of states depends on energy differential conductance is non-linear with voltage

Tunneling electrons into Tomonaga-Luttinger liquids

X.G. Wen (1990)

periphery deformation of 1/3 incompressible FQHE electron liquid

X

Y

DvFY

),( tXy

Xyns

~

1.~

(excess charge density / length)

nns

3/1

Classical hydrodynamics

+ field quantization:

)'()(~

),(~ XXiXX

e+ electron creation operator on the edge

+ Fermi statistics :

12

1 Fermions

s

Tunneling into Chiral Luttinger liquid (FQHE regime)

X.G. Wen (1990)

periphery deformation of 1/3 incompressible FQHE electron liquid

X

Y

DvFY

),( tXy

nns

3/1

Classical hydrodynamics

Xyns

~

1.~

(excess charge density / length)

+ field quantization:

)'()(~

),(~ XXiXX

e+ electron creation operator on the edge

+ Fermi statistics :

properties of a Luttinger liquid with g =

Tunneling into Chiral Luttinger liquid (FQHE regime)

n+ GaAs2 DEG

V

e

3/1

observed7.2

predicted3

also observed :

Tk

eVfT

dV

dIVT

dV

dI

B2)0,(),(

(voltage and temperature play the same role)

power law variationof the current / voltage

Chiral-Luttinger prediction:

A.M. Chang (1996)

tunneling from a metal to a FQHE edge

Simplest theory predicts for

)12/( pp

3/1

power laws are stille observed as expectedbut 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)

tunneling from a metal to a FQHE edge

GaAs

GaAlAsSi+

heterojunction

2D electrons

Atomically controlled epitaxial growth GaAs/Ga(Al)As heterojunction

CLEAN 2D electron gas

constriction(Quantum Point Contact)

200nm

(top view )

100 nm

(edge channel)

tunneling between FQHE edges

energy

high barrier3/1 3/1V

dV

dI

4

)11

(2

(doubled)

energy

weak barrier3/1 3/1

low energy

4VdV

dI

large energy

3/4

2

3

VdV

dI

IVh

eI

B

B

even the weakest barrierleads to strong reflection

at low energy !


'),(

')1(ln

)(2cosh

1),(

'0

deV V

'

),(2

),(2

'0 )1()1(ln)(2cosh

1),(

'0

'0

deeV

VT

VV

T

V

de

e

TTh

TeTVTI

VT

V

VT

V

B

B

),(2

),(2

2 '0

'0

1

1ln

)/(ln(2cosh

1),,(

0kink / anti-kink (charged solitons ) in the phase field (x,t)

breather (neutral soliton )

thermodynamic Bethe Ansatz self consistent equations

Expression of the current

… similar calculation for shot noise

folded into:

x

matters only

conserved is

: and outgoing

R

R

L

L

RL

)(L

)(R

)0()0(cos RLBH

P. Fendley, A. W. W. Ludwig, and H. Saleur,Phys. Rev. Lett. 74, 3005 (1995); 75, 2196 (1995);

tunneling between FQHE edges (TBA solution of the B.Sine-Gordon model)

(impurity, strength TB )

eV << TB

)1(22

)...()(

V

T

h

eVG B

)11

(2

)...()(

BT

VVG

eV >> TB

Numerical calculation of G(V) using the exact solution by FLS (1996) (P.Roche + C. Glattli 2002 )

Tk

eV

T

TG

dV

dI

BB 2,

dV

dI


BT

BT

TVG ,0

0,

BT

TVG

0

TV

TB

or fixed at

varying

energy

very weak barrier

3/1 3/1

tunneling between FQHE edges : experimental comparison

-150 -100 -50 0 50 100 1500,0

0,2

0,4

0,6

0,8

1,0

Vg = 32.22 mV

Vg = 23.24 mV

Vg = 17.55 mV

Vg = 12.23 mV

Vg = 7.77 mV

Vg = 2.35 mV

dif

fere

nti

al co

nd

ucta

nce (

e2 /

3 h

)

bias voltage V (V)

-3 -2 -1 0 1 2 3 40

2

4

6

8

10

12

14

16

18

20

22

24

Vg = 2.07 mV

eV / (2kBT)

dI/d

V (V

,T)

/ d

I/d

V (0

,T)

47.1 mK36.6 mK32.4 mK25.6 mK19.7 mK

0,1 10,01

0,1

1

10

e V / ( 2 kBT )

dI/d

V (

V,T

) / d

I/d

V (

O,T

) -

1

19.7 mK25.6 mK32.4 mK36.6 mK47.1 mK

(a)

(b)

Tk

eV

T

TG

dV

dI

BB 2,

1

3/1

32

4

3/1

4

10/10

10/

GGVG

GGVG

for

for

scaling V/T is OK

… but dI/dV varies as the secondinstead of the fourth power of V( or T)predicted by perturbative renormalizationapproach.

solid line: renormalization fixed point limit

tunneling between FQHE edges : experimental comparison

0 5 10 15 20

0,00,10,20,30,40,50,60,70,80,91,01,1

low temperature

high temperature

diffe

ren

tial c

ond

ucta

nce

(e 2/ 3

h )

e V / 2 k B

T

Finite temperature calculation using the TBA solution of the boundary Sine-Gordon model

weak barrier

3/1 3/1

Tk

eVfT

dV

dIVT

dV

dI

B2)0,(),(

(Saclay 2000)

to observe exponent =4 one needs very low temperature and conductance 10-4 X e2/3h !

(scaling law experimentally observed (Saclay 1998) )

0 5 10 15 20

0,00,10,20,30,40,50,60,70,80,91,01,1

low temperature

high temperature

diffe

ren

tial c

ond

ucta

nce

(e 2/ 3

h )

e V / 2 k B

T

weak barrier

3/1 3/1

Tk

eVfT

dV

dIVT

dV

dI

B2)0,(),(

(Saclay 2000)

(scaling law experimentally observed (Saclay 1998) )

e/3

e

Finite temperature calculation using the Fendley, Ludwig, Saleur (1995) exact solution

to observe exponent =4 one needs very low temperature and conductance 10-4 X e2/3h !

metal =1/3 e

e/3e/3

) liquid Luttinger:later also (see

char. V-Ilinear -non)( 3VVI

IffSI I 2)0(2 qq = e/3

e/3=1/3 =1/3

-e/3

e/3

e

e/3

Laughlin quasiparticleson the edge

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)

probing quasiparticles via tunneling experiments,two different approaches:

)( D

)(Lf

)(Rf

eV)1( D

eV

ht

0I

( i ) ( t )

( r )

)(tI

incoming current :

heVeI /0

(noiseless thanks to Fermi statistics)

transmitted current :

Vh

eDIDI

2

0

)1(2)1(2 02 DfIeDDfIeI

current noise in B.W. f :

Variance of partioning binomial statistics

ring)backscatte (strong

charge electron eq

qII

II O

22

ring)backscatte (weak

charge hole eq

qII

IIII OOB

22

2 limiting cases:

The binomial statistics of Shot Noise (no interactions)

quantum point contact (B=0)

first mode :slope ~ (1 - D1 )

B

I

kG

ST

4*

Kumar et al. PRL (1996)

0,0

0,2

0,4

0,6

0,8

1,0

1 1 T

T TT

2 2

2

11

Fa

no

re

du

ctio

n f

act

or

Conductance 2e² / h

0. 0.5 1. 1.5 2. 2.5

1

.8

.6

.4

.2

0 M. I. Reznikov et al., Phys. Rev. Lett. 75 (1995) 3340.

A. Kumar et al. Phys. Rev. Lett. 76 (1996) 2778..

eIS

D

I 2.F

1F

-50 0 50 1000

1

2

3

Vg ( mV )

co

ndu

ctan

ce

( 2e

2 / h

)

n

nDh

eG .

2 2

nm70Fm2010. elastl

(ballistic conductor)

2-Delectron

gas

Gate

Gate

(Saclay 1996)

e

e

strong barrier :

= 1 = 1V

)(tIhVeI /2

0

hVeI /2

0

)(tI B

0II

1 2 DeISI

1 2 DeIS BI

)1(2 0 DDIeS I

BIII

hVeI

0

20 /

transmitted (D) reflected (1-D)

(rarely transmitted electrons)

(incoming electrons)

e e

(rarely transmitted holes)

weak barrier :

Shot Noise in IQHE regime

e

e/3

strong barrier :

= 1/3 = 1/3V

)(tIhVeI 3/20

hVeI 3/20

)(tI B

0II

1 2 DeISI

1 3

2 DIe

S BI

BIII

hVeI

0

20 3/

transmitted (D) reflected (1-D)

(rarely transmitted electrons)

(incoming electrons)

e e

(rarely transmitted holes)

weak barrier :

e e

e/3 e/3

Shot Noise in IQHE regime

e / 3

Direct evidence of fractional charge

)3/(2. fIeII RR

L. Saminadayar et al. PRL (1997).De Picciotto et al. Nature (1997)

charge q=e/3

measure of the anti-correlated transmitted X reflectedcurrent fluctuations (electronic Hanbury-Brown Twiss)

charge q=e

-150 -100 -50 0 50 100 1500,0

0,2

0,4

0,6

0,8

1,0

Vg = 32.22 mV

Vg = 23.24 mV

Vg = 17.55 mV

Vg = 12.23 mV

Vg = 7.77 mV

Vg = 2.35 mV

dif

fere

nti

al c

on

du

ctan

ce (

e2 /

3 h

)

bias voltage V (V)

-0,3 -0,2 -0,1 0,0 0,1 0,2 0,30,12

0,14

0,16

0,18

0,20

0,22

0,24

VG = 23.5 mV

Tvraie

= 40 mK

theory : 2 e/3 Ib coth (e/3V

ds/2kT)

data

curr

en

t no

ise

(1

0 -

28 A

2 /

Hz)

2 e/3 IB (10 -28 A 2 / Hz)

charge e

??

charge e/3

V. Rodriguez et al (2000)

From fractional to integer charges

-30 -20 -10 0 10 20 300,0

0,1

0,2

0,3

0,4

0,5 T = 47.8 mK T = 36.7 mK

Vg = 2.07 mV

diffe

rent

ial c

ondu

ctan

ce (

e 2

/ 3h

)

Bias Voltage ( V )

-30 -20 -10 0 10 20 30

-0,1

0,0

0,1

Vg = 2.07 mV

T = 47.8 mK T = 36.7 mK

Bias Voltage ( V )

curr

ent

( nA

)

-30 -20 -10 0 10 20 30

0,0

0,1

0,2

0,3

0,4 T = 47.8 mK T = 36 mK

exce

ss c

urre

nt n

oise

(10

- 2

8 A 2

/ H

z)

Bias Voltage ( V )

-0,3 -0,2 -0,1 0,0 0,1 0,2 0,3

0,0

0,1

0,2

0,3

exce

ss c

urre

nt n

oise

(10

- 2

8 A 2

/ H

z)

2 e I ( 10 -28

A 2 / Hz )

(a)

(d)(b)

(c)

e charge for

noiseSchottky

IeS I 2

From fractional to integer charges

0,0 0,4 0,8 1,2 1,6 2,0 2,40,0

0,4

0,8

1,2

1,6

2,0

2,4

G(V=0) = 0.97 G1/3

G(V=0) = 0.44 G1/3

G(V=0) = 0.2 G1/3

G(V=0) = 0.07 G1/3

G(V=0) = 0.016 G1/3

SI /

G0k

BT

{2e*IBT Cotanh(e*V/2k

B) + 4 T 2G

1/3T } / G

0k

B

0 2 4 60,0

0,4

0,8

1,2

1,6

2,0

2,4

eV / (2kB)

SI /

G0k

BT

2*

*

3/1

22

coth05.0 Tkk

VeTIeS

G

GB

B

BI

exactsolution(Bethe Ansatz)

dotted line:empirical binomial noise formula for backscatterede/3 quasiparticles

numerical calculation of the finite temperatureshot noise

3/*/*

/ee

hee

dVdIT

with

extremely good !

P. Fendley and H. Saleur, Phys. Rev. B 54, 10845 (1996)

(P.Roche + C. Glattli 2002 )

222

coth)1( Tkk

VeTIeS B

BI

eehdVdIT

05.03/1

G

G

2*

*

3/1

22

coth05.0 Tkk

VeTIeS

G

GB

B

BI

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





graphene energy band structure

Luttinger Liquid effects in Single Wall Nanotubes

e

Electron tunneling into a SWNT excites 1D plasmons in the nanotubes

giving rise to Luttinger liquid effects

plasmon

plasmon

SWNT

TTG

VVG

)(

)(Non-linear conductance:

provided kT or eV < hvF / L


L

)or (ds

eVTkL

vhB

F

Observation of LL effects requires

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):

e V

I 4/21

gg

VdVdI

Luttinger-Liquid behavior in Crossed Metallic Single-Wall Nanotubes

differential tube-tube conductance

~700 nm

g = 0.16

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, 216804 (2004)

OUTLINE





Possible future experimental investigations

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

see arXiv: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 ?

K. Bennaceur (Saclay SPEC)

holes electrons

-60 -40 -20 0 20 40 600

200

400

600

800

(2 X 7.63 GHz)

(2 X 4.22 GHz)

TN

oise

( K

ref

ered

to 5

0

)

VDrain-Source

(µV)

7.63GHz 4.22GHz

E. Zakka-Bajani PRL 2007

R12,42

ens-paris

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