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 - … ) (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. 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 effect Edges as Chiral LL: Carbone Nanotubes signatures of T-LL: (on going or foreseen experimental projects)

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

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 )

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 !!!

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

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

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

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

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 :

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

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)

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)

(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)

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 !

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

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 )

tunneling between FQHE edges : experimental comparison energy very weak barrier

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

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

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

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:

(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 0. 0.5 1. 1.5 2. 2.5 .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) 2778..

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

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

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)

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

From fractional to integer charges

extremely good ! 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 effect Edges 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

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 ~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

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, 216804 (2004)

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

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 ? E. Zakka-Bajani PRL 2007 R12,42 electrons holes K. Bennaceur (Saclay SPEC)