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
  g0.2 to 0.3
• Egger Gogolin, PRL 1997
• Kane, Balents Fisher, PRL1997
• Experiment Luttinger power-law conductance
  through weak link, gives g0.22
  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

• Ngt2 branches, junction with S matrix
• implies wavefunction matching at node

Crossover from full to no transmission tuned by ?
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 g1/2

• Nonlinear conductance
• with

18
Nonlinear conductance
g1/2
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Ideal junction Fixed point
g1/3

• 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 g3/8 (Toulouse limit
  in suitable rotated picture)
• Nonperturbative crossover from truly insulating
  node to disconnected tube 1 perfect wire 23

21
Asymmetric Y junction g3/8

• 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
g1/4, T0
1) Perfect zero-bias anomaly 2) Dips are turned
into peaks for finite cross voltage, with
new minima
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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

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Coulomb drag Transconductance

• Strictly local coupling Linear transconduc-tance
  always vanishes
• Finite length Couplings in /- sectors differ

Now nonzero linear transconductance, except at
T0!
28
Linear transconductance g1/4
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 T0 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, glt1/2
• 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 s Model
• Equivalent to Coulomb Blockade spectral density
  I(?) 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

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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