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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
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
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
Conductance scaling
Conductance across kink:
Universal scaling of nonlinear conductance:
r.h.s. is only function of V/T
22.02/)1(, 1 ggTG
Tk
ieV
Tk
eV
Tk
ieV
Tk
eVdVdIT
BB
BB
221Im
2
1
2coth
221
2sinh/
2
Evidence for Luttinger liquid
Yao et al., Nature 1999
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
Multi-terminal circuits: Crossed tubes
Fuhrer et al., Science 2000Terrones et al., PRL 2002
Fusion: Electron beam welding(transmission electron microscope)
By chance…
Nanotube Y junctions
Li et al., Nature 1999
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
Coupling to voltage reservoirs
Two-terminal case, applied voltage
Left/right reservoir injects `bare´ density of R/L moving charges
Screening: actual charge density isFRL
FLR
vL
vL
2/)2/(
2/)2/(0
0
RLeU
)()( 002LRLR gx
Egger & Grabert, PRL 1997
Radiative boundary conditions
Difference of R/L currents unaffected by screening:
Solve for injected densities
boundary conditions for chiral density near adiabatic contacts
)()()()( 00 xxxx LRLR
FLR v
eUL
gL
g
2)2/(1
1)2/(1
122
Egger & Grabert, PRB 1998Safi, EPJB 1999
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
Description of junction (node) ?
Landauer-Büttiker: Incoming and outgoing states related via scattering matrix
Difficult to handle for correlated systems What to do ?
Chen, Trauzettel & Egger, PRL 2002Egger, Trauzettel, Chen & Siano, cond-mat/0305644
)0()0( inout S
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
Ideal symmetric junctions N>2 branches, junction with S matrix
implies wavefunction matching at node
1...
............
...1
...1
zzz
zzz
zzz
S
0,2
iN
z
)0(...)0()0( 21 N
)0()0()0( ,, outjinjj
Crossover from full to no transmission tuned by λ
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
)0(...)0(1 N
j
iij
I
h
eG
0i
iI
Solution for Y junction with g=1/2Nonlinear conductance:
with
T
VUieT
T
eV iiB
B
i
22/
2
1Im
2
2
22
0
)1/(10
)2(
22),(
/
NN
NNNw
wDT gB
ij j
j
i
iii U
VU
VG 19
21
9
8
Nonlinear conductance
g=1/2
F
F eU
32
1
Ideal junction: Fixed point
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 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 Nonperturbative solution:
Asymmetry contribution
Strong asymmetry limit:
2/, 03,23,2
011 IIIIII
DWW
T
eIIiWeWI
B
BB
/
2
/2/2
2
1Im
2
01
2/,0 01
03,23,21 IIII
Crossed tubes: Local Coulomb drag Different limit: Weakly coupled crossed
nanotubes Single-electron tunneling between tubes irrelevant Electrostatic coupling relevant for strong
interactions, Without tunneling:
Local Coulomb drag
2/1g
Komnik & Egger, PRL 1998, EPJB 2001
Hamiltonian for crossed tubes Without tunneling:
Rotated boson fields:
Boundary condition decouples: Hamiltonian also decouples!
22
2121
)(2
)0(4cos)0(4cos
iiFi
Lutt
LuttLutt
dxg
vH
ggcHHH
2/)( 21 UUU
2/)()()( 21 xxx
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?
)0(8cos
gcHH
HHH
Lutt
gg 2
Crossed tubes: Conductance
g=1/4, T=0
11G
1) Perfect zero-bias anomaly2) Dips are turned into peaks for finite cross voltage, with new minima
Experiment: Crossed nanotubes Measure nonlinear
conductance for cross voltage
Zero-bias anomaly for small cross voltage
Conductance dip becomes peak for larger cross voltage
11G
Kim et al., J. Phys. Soc. Jpn. 2001
)(1 meVU
meVUmeV 2020 2
Coulomb drag: Transconductance Strictly local coupling: Linear transconduc-
tance always vanishes Finite length: Couplings in +/- sectors differ
21G
BB
g
B
L
L
FF
TT
DcDT
xkkdxL
ccc
)21/(1
2/
2/
2,1,
/
)(2cos
Now nonzero linear transconductance,except at T=0!
Linear transconductance: g=1/4
TTd
dd
ddG
B 2/)2/1(1
)2/1(1
2
1'
'
21
1BT
Absolute Coulomb drag
For long contact & low temperature: Transconductance approaches maximal value
At zero temperature, linear drag conductance vanishes (in not too long contact)
2/1)0/,0(21 BB TTTG
Averin & Nazarov, PRL 1998Flensberg, PRL 1998Komnik & Egger, PRL 1998, EPJB 2001
0)0(21 TG
Coulomb drag shot noise
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)
)0()()( ItIdteP ti
)(2 UeIP BS
00,0 121 PUU
Trauzettel, Egger & Grabert, PRL 2002
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
0)0()( ItI
2/)(21 PPPP
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
RR 10...
Fv
E
/
1
MWNTs: Ballistic limit
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
Egger, PRL 1999
N1
Experiment: TDOS of MWNT
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
bulkend 2
Basel group, PRL 2001
Geometry dependence
Interplay of disorder and interaction Coulomb interaction enhanced by disorder Microscopic nonperturbative theory:
Interacting Nonlinear σ Model Equivalent to Coulomb Blockade: spectral
density I(ω) of intrinsic electromagnetic modes
Egger & Gogolin, PRL 2001, Chem. Phys. 2002Rollbühler & Grabert, PRL 2001
1)(),0(
expRe)(
0
0
tieId
tTJ
tJiEtdt
EP
Intrinsic Coulomb blockade TDOS Debye-Waller factor P(E):
For constant spectral density: Power law with exponent Here:
Tk
TkE
B
B
e
eEPd
E/
/
0 1
1)(
)0( I
2/,1/
/Re)(2
)(
200
*
*2/1
2
2**0
F
n
vDUDD
DDR
nDiDD
UI
Field/charge diffusion constant
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
DDD
R/ln
2*
0
2)2/( RDEE Thouless
R
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
1/,12/,10 0 RvvUR FF
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