Transport through junctions of interacting quantum wires and nanotubes

1. Transport through junctions of interacting quantum wires and nanotubes R. Egger Institut für Theoretische Physik Heinrich-Heine Universität Düsseldorf S. Chen, S. Gogolin, H. Grabert, A. Komnik, H. Saleur, F. Siano, B. Trauzettel

2. Overview • Introduction: Luttinger liquid in nanotubes • Multi-terminal circuits • Landauer-Büttiker theory for junction of interacting quantum wires • Local Coulomb drag: Conductance and perfect shot noise locking • Multi-wall nanotubes • Conclusions and outlook

3. Single-wall carbon nanotubes • Prediction: SWNT is a Luttinger liquid with g~0.2 to 0.3 Egger & Gogolin, PRL 1997 Kane, Balents & Fisher, PRL1997 • Experiment: Luttinger power-law conductance through weak link, gives g~0.22 Yao et al., Nature 1999 Bockrath et al., Nature 1999

4. Conductance scaling • Conductance across kink: • Universal scaling of nonlinear conductance: r.h.s. is only function of V/T

5. Evidence for Luttinger liquid Yao et al., Nature 1999

6. Luttinger liquid properties • Momentum distribution: no jump at Fermi surface, power-law scaling • Tunneling density of states power-law suppressed, with different end/bulk exponent • Spin-charge separation • Fractional charge and statistics • Networks of nanotubes: Experiment? Theory? Dekker group, Delft

7. Multi-terminal circuits: Crossed tubes Fusion: Electron beam welding (transmission electron microscope) By chance… Fuhrer et al., Science 2000 Terrones et al., PRL 2002

8. Nanotube Y junctions Li et al., Nature 1999

9. Landauer-Büttiker theory ? • Standard scattering approach useless: • Elementary excitations are fractionalized quasiparticles, not electrons • No simple scattering of electrons, neither at junction nor at contact to reservoirs • Generalization to Luttinger liquids • Coupling to reservoirs via radiative boundary conditions • Junction: Boundary condition plus impurities

10. Coupling to voltage reservoirs • Two-terminal case, applied voltage • Left/right reservoir injects `bare´ density of R/L moving charges • Screening: actual charge density is Egger & Grabert, PRL 1997

11. Radiative boundary conditions Egger & Grabert, PRB 1998 Safi, EPJB 1999 • Difference of R/L currents unaffected by screening: • Solve for injected densities boundary conditions for chiral density near adiabatic contacts

12. Radiative boundary conditions … • hold for arbitrary correlations and disorder in Luttinger liquid • imposed in stationary state • apply to multi-terminal geometries • preserve integrability, full two-terminal transport problem solvable by thermodynamic Bethe ansatz Egger, Grabert, Koutouza, Saleur & Siano, PRL 2000

13. Description of junction (node) ? Chen, Trauzettel & Egger, PRL 2002 Egger, Trauzettel, Chen & Siano, cond-mat/0305644 • Landauer-Büttiker: Incoming and outgoing states related via scattering matrix • Difficult to handle for correlated systems • What to do ?

14. Some recent proposals … • Perturbation theory in interactions Lal, Rao & Sen, PRB 2002 • Perturbation theory for almost no transmission Safi, Devillard & Martin, PRL 2001 • Node as island Nayak, Fisher, Ludwig & Lin, PRB 1999 • Node as ring Chamon, Oshikawa & Affleck, cond-mat/0305121 • Our approach: Node boundary condition for ideal symmetric junction (exactly solvable) • additional impurities generate arbitrary S matrices, no conceptual problem Chen, Trauzettel & Egger, PRL 2002

15. Ideal symmetric junctions • N>2 branches, junction with S matrix • implies wavefunction matching at node Crossover from full to no transmission tuned by λ

16. Boundary conditions at the node • Wavefunction matching implies density matching can be handled for Luttinger liquid • Additional constraints: • Kirchhoff node rule • Gauge invariance • Nonlinear conductance matrix can then be computed exactly for arbitrary parameters

17. Solution for Y junction with g=1/2 Nonlinear conductance: with

18. Nonlinear conductance g=1/2

19. Ideal junction: Fixed point g=1/3 • Symmetric system breaks up into disconnected wires at low energies • Only stable fixed point • Typical Luttinger power law for all conductance coefficients • Solvable for arbitrary correlations

20. Asymmetric Y junction • Add one impurity of strength W in tube 1 close to node • Exact solution possible for g=3/8 (Toulouse limit in suitable rotated picture) • Nonperturbative crossover from truly insulating node to disconnected tube 1 + perfect wire 2+3

21. Asymmetric Y junction: g=3/8 • Nonperturbative solution: • Asymmetry contribution • Strong asymmetry limit:

22. Crossed tubes: Local Coulomb drag Komnik & Egger, PRL 1998, EPJB 2001 • Different limit: Weakly coupled crossed nanotubes • Single-electron tunneling between tubes irrelevant • Electrostatic coupling relevant for strong interactions, • Without tunneling: Local Coulomb drag

23. Hamiltonian for crossed tubes • Without tunneling: • Rotated boson fields: • Boundary condition decouples: • Hamiltonian also decouples!

24. Map to decoupled 2-terminal models • Two effective two-terminal (single impurity) problems for • Take over exact solution for two-terminal problem • Dependence of current on cross voltage?

25. Crossed tubes: Conductance g=1/4, T=0 1) Perfect zero-bias anomaly 2) Dips are turned into peaks for finite cross voltage, with new minima

26. Experiment: Crossed nanotubes Kim et al., J. Phys. Soc. Jpn. 2001 • Measure nonlinear conductance for cross voltage • Zero-bias anomaly for small cross voltage • Conductance dip becomes peak for larger cross voltage

27. Coulomb drag: Transconductance • Strictly local coupling: Linear transconduc-tance always vanishes • Finite length: Couplings in +/- sectors differ Now nonzero linear transconductance, except at T=0!

28. Linear transconductance: g=1/4

29. Absolute Coulomb drag Averin & Nazarov, PRL 1998 Flensberg, PRL 1998 Komnik & Egger, PRL 1998, EPJB 2001 • For long contact & low temperature: Transconductance approaches maximal value • At zero temperature, linear drag conductance vanishes (in not too long contact)

30. Coulomb drag shot noise Trauzettel, Egger & Grabert, PRL 2002 • Shot noise at T=0 gives important information beyond conductance • For two-terminal setup, one weak impurity, DC shot noise carries no information about fractional charge • Crossed nanotubes: For must be due to cross voltage (drag noise)

31. Shot noise transmitted to other tube ? • Mapping to decoupled two-terminal problems implies • Consequence: Perfect shot noise locking • Noise in tube 1 due to cross voltage, exactly equal to noise in tube 2 • Requires strong interactions, g<1/2 • Effect survives thermal fluctuations

32. Multi-wall nanotubes: Luttinger liquid? • Russian doll structure, electronic transport in MWNTs usually in outermost shell only • Typically 10 transport bands due to doping • Inner shells can create `disorder´ • Experiments indicate mean free path • Ballistic behavior on energy scales

33. MWNTs: Ballistic limit Egger, PRL 1999 • Long-range tail of interaction unscreened • Luttinger liquid survives in ballistic limit, but Luttinger exponents are closer to Fermi liquid, e.g. • End/bulk tunneling exponents are at least one order smaller than in SWNTs • Weak backscattering corrections to conductance suppressed as 1/N

34. Experiment: TDOS of MWNT Bachtold et al., PRL 2001 (Basel group) • DOS observed from conductance through tunnel contact • Power law zero-bias anomalies • Scaling properties similar to a Luttinger liquid, but: exponent larger than expected from Luttinger theory

35. Tunneling density of states: MWNT Basel group, PRL 2001 Geometry dependence

36. Interplay of disorder and interaction Egger & Gogolin, PRL 2001, Chem. Phys. 2002 Rollbühler & Grabert, PRL 2001 • Coulomb interaction enhanced by disorder • Microscopic nonperturbative theory: Interacting Nonlinear σ Model • Equivalent to Coulomb Blockade: spectral density I(ω) of intrinsic electromagnetic modes

37. Intrinsic Coulomb blockade • TDOS Debye-Waller factor P(E): • For constant spectral density: Power law with exponent Here: Field/charge diffusion constant

38. Dirty MWNT • High energies: • Summation can be converted to integral, yields constant spectral density, hence power law TDOS with • Tunneling into interacting diffusive 2D metal • Altshuler-Aronov law exponentiates into power law. But: restricted to

39. Numerical solution • Power law well below Thouless scale • Smaller exponent for weaker interactions, only weak dependence on mean free path • 1D pseudogap at very low energies

40. Conclusions • Luttinger liquid behavior in SWNTs offers new perspectives: Multi-terminal circuits • Theory beyond Landauer-Büttiker • New fixed points: Broken-up wires, disconnected branches • Coulomb drag: Absolute drag, noise locking • Multi-wall nanotubes: Interplay disorder-interactions