Transport through junctions of interacting quantum wires and

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

Overview n n n 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

Single-wall carbon nanotubes n Prediction: SWNT is a Luttinger liquid with g~0. 2 to 0. 3 Egger & Gogolin, PRL 1997 Kane, Balents & Fisher, PRL 1997 n Experiment: Luttinger power-law conductance through weak link, gives g~0. 22 Yao et al. , Nature 1999 Bockrath et al. , Nature 1999

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

Evidence for Luttinger liquid Yao et al. , Nature 1999

Luttinger liquid properties n n n 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

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

Nanotube Y junctions Li et al. , Nature 1999

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

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

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

Radiative boundary conditions … n n 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

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

Some recent proposals … n Perturbation theory in interactions Lal, Rao & Sen, PRB 2002 n n 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) q additional impurities generate arbitrary S matrices, no conceptual problem Chen, Trauzettel & Egger, PRL 2002

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

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

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

Nonlinear conductance g=1/2

Ideal junction: Fixed point n n 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 g=1/3

Asymmetric Y junction n 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

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

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

Hamiltonian for crossed tubes n Without tunneling: n Rotated boson fields: n Boundary condition decouples: Hamiltonian also decouples! n

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

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

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

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

Linear transconductance: g=1/4

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

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

Shot noise transmitted to other tube ? n Mapping to decoupled two-terminal problems implies n Consequence: Perfect shot noise locking q q q 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

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

MWNTs: Ballistic limit Egger, PRL 1999 n n 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

Experiment: TDOS of MWNT n n n 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 Bachtold et al. , PRL 2001 (Basel group)

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

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

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

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

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

Conclusions n n n 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 disorderinteractions