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Electronic transport in one-dimensional wires

Electronic transport in one-dimensional wires. Akira Furusaki (RIKEN). Outline. Tomonaga-Luttinger (TL) liquid Bosonization Single impurity in a TL liquid Two impurities in a TL liquid linear conductance G Random-matrix approach to transport in disordered wires.

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Electronic transport in one-dimensional wires

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  1. Electronic transportin one-dimensional wires Akira Furusaki (RIKEN)

  2. Outline • Tomonaga-Luttinger (TL) liquid • Bosonization • Single impurity in a TL liquid • Two impurities in a TL liquidlinear conductanceG • Random-matrix approach to transport in disordered wires Electronic transport in 1D wires

  3. 1D metals= Tomanaga-Luttinger liquid • No single-particle excitations • Collective bosonic excitationsspin-charge separation charge density fluctuations spin density fluctuations • Power-law decay of correlation functions (T=0) tunneling density of states Electronic transport in 1D wires

  4. TL liquids are realized in: • Very narrow (single-channel) quantum wires • edge states of fractional quantum Hall liquids • Carbon nanotubes Electronic transport in 1D wires

  5. Interacting spinless fermions • Simplified continuum model kinetic energy short-range repulsive interaction (forward scattering) Electronic transport in 1D wires

  6. Abelian Bosonization • Fermions = Bosons in 1D Electronic transport in 1D wires

  7. Electron density Electronic transport in 1D wires

  8. Kinetic energy Electronic transport in 1D wires

  9. Bosonized Hamiltonian TL liquid parameter g g < 1: repulsive interaction FQHE edgeg = 1: non-interacting case g > 1: attractive interaction Interacting fermions = free bosons Electronic transport in 1D wires

  10. Correlation functions（T=0） Scaling dimension of is Electronic transport in 1D wires

  11. Single impurity • Non-interacting case (free spinless fermions) transmission probability Electronic transport in 1D wires

  12. Current Conductance G changes continuously. no temperature dependence. is a marginal perturbation Electronic transport in 1D wires

  13. Interacting spinless fermions reflection at the barrier potential Hamiltonian free boson + = pinning of charge density wave electric current Electronic transport in 1D wires

  14. Partition function (path integral) effective action for linear: dissipation due to gapless excitations in TL liquid (Caldeira-Leggett: Macroscopic Quantum Coherence) a particle (with coordinate ) moving in a cosine potential with friction Electronic transport in 1D wires

  15. Renormalization-group analysis • Weak-potential limit weak perturbation: scaling equation (lowest order): renormalized potential: conductance Electronic transport in 1D wires

  16. Strong-potential limit (weak-tunneling limit) duality transformation[A. Schmid (’83); compact QED by A.M. Polyakov] 　　　“dilute instanton (=tunneling) gas” t: tunneling matrix element (fugacity) Electronic transport in 1D wires

  17. scaling equation: renormalized tunneling matrix element: conductance Electronic transport in 1D wires

  18. Flow diagram for transmission probability(Kane & Fisher, 1992) 1 Trans. Prob. g<1 (repulsive int.) perfect reflection at T=0 g=1 (free fermions) marginal g>1 (attractive int.) perfect transmission at T=0 0 g 1 Electronic transport in 1D wires

  19. Exact results • “Toulouse limit” g=1/2 introduce new fields refermionization quadratic Hamiltonian cf. 2-channel Kondo problem (Emery-Kivelson, 1992) Electronic transport in 1D wires

  20. Conductance at g=1/2 • General gTheboundary sine-Gordon theory is exactly solvable(Ghoshal & Zamolodchikov, 1994) Bethe ansatz elastic single-quasiparticle S-matrix(Fendley, Ludwig & Saleur, 1995) Electronic transport in 1D wires

  21. Spinful case (electrons)(Furusaki & Nagaosa, 1993; Kane & Fisher, 1992) charge boson: spin boson: Hamiltonian : non-interacting electrons : repulsive interactions : if spin sector has SU(2) symmetry Electronic transport in 1D wires

  22. Weak-potential limit • Strong-potential limit (weak-tunneling limit)single-electron tunneling: t • RG flow diagram critical surface at intermediate coupling 1 1 Trans. Prob. Trans. Prob. 0 0 1 Electronic transport in 1D wires

  23. External leads (Fermi-liquid reservoir)(Maslov & Stone, 1994) Tomonaga-Luttinger liquid: Fermi-liquid leads: Action Current Ivs Electric field E dc conductance is not renormalized by the e-e interaction if the wire is connected to Fermi-liquid reservoirs Electronic transport in 1D wires

  24. Weak e-e interactions (Matveev, Yue & Glazman, 1993) small parameter: V(q): Fourier transform of interaction potential scaling equation for the transmission probability lowest order in but exact in conductance Electronic transport in 1D wires

  25. Coulomb interactions (Nagaosa & Furusaki, 1994; Fabrizio, Gogolin & Scheidel, 1994) : width of a quantum wire scaling equation for tunneling conductance stronger suppression than power law Electronic transport in 1D wires

  26. Experiments on tunneling • Edge states in FQHE(Chang, Pfeiffer & West, 1996) tunneling between a Fermi liquid and edge state [Fig. 1 & Fig. 2 of PRL 77, 2538 (1996) were shown in the lecture] Electronic transport in 1D wires

  27. Single-wall carbon nanotubesYao, Postma, Balents & Dekker, Nature 402, 273 (1999) [Fig. 1 and Fig. 3 were shown in the lecture.] Segment I & II: bulk tunneling Across the kink: end-to-end tunneling exp: Electronic transport in 1D wires

  28. Resonant Tunneling (Double barriers) L R • Non-interacting casetransmission amplitude: t has maximum whenresonance (symmetric barrier) symmetric case backscattering is irrelevant asymmetric case backscattering is marginal = single impurity x 0 d Electronic transport in 1D wires

  29. When life time of discrete levels Conductance if coherent tunneling if incoherent sequential tunneling peak width Electronic transport in 1D wires

  30. Resonant tunneling in TL liquids Spinless fermions Hamiltonian gate voltage Current Excess charge in [0, d ] is massive Electronic transport in 1D wires

  31. Weak-potential limit(Kane & Fisher, 1992)effective action for single-barrier problemscaling equation if (symmetric) and (on resonance) 1 g g 1/4 Electronic transport in 1D wires

  32. Resonance line shape symmetric ¼ < g < 1is the only relevant operator, on resonanceuniversal line shapepeak width not Lorentzian Electronic transport in 1D wires

  33. Weak-tunneling limit(Furusaki & Nagaosa, 1993; Furusaki,1998) • Off resonance process is not allowed at low T virtual tunneling • On resonancesequential tunnelinglife time due to tunneling through a barrierpeak width Electronic transport in 1D wires

  34. 1 0 1 Phase diagram at T=0 1 • Symmetric barriers • Asymmetric barriersg<1 g=1 g>1 Transmission probability g 0 1/4 1/2 1 Electronic transport in 1D wires

  35. T > 0 • Weak potential • Weak tunnelingsequential tunneling Electronic transport in 1D wires

  36. Experiments on resonant tunneling in TL liquids Auslaender et al., Phys. Rev. Lett. 84, 1764 (2000) Electronic transport in 1D wires

  37. Carbon nanotubes Postma et al., Science 293, 76 (2001) Electronic transport in 1D wires

  38. Summary • In 1D e-e interaction is crucial Tomonaga-Luttinger liquid • Repulsive e-e interaction backward potential scattering is relevant power-law suppression of tunnel density of states • Problems • nontrivial fixed points at intermediate coupling • Resonant-tunneling experiment? Electronic transport in 1D wires

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