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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

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  1. 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)

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

  3. 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)

  4. (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

  5. 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

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

  7. 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

  8. 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

  9. 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

  10. + 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)

  11. (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

  12. 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)

  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. 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)

  15. 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

  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. (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 )

  18. tunneling between FQHE edges : experimental comparison energy 0 very weak barrier

  19. 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. 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

  23. ( 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

  24. 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..

  25. 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

  26. 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

  27. 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)

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

  29. From fractional to integer charges

  30. 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 !

  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, 233301 (2004)

  32. 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

  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. ~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

  39. 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)

  40. 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

  41. 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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