Transcript Transport through interacting quantum wires and nanotubes

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
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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
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Prediction: SWNT is a
Luttinger liquid with
g~0.2 to 0.3
Egger & Gogolin, PRL 1997
Kane, Balents & Fisher, PRL1997
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Experiment: Luttinger
power-law conductance
through weak link,
gives g~0.22
Yao et al., Nature 1999
Bockrath et al., Nature 1999
Conductance scaling
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Conductance across kink:
G  T  ,  ( g 1 1) / 2  g  0.22
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Universal scaling of nonlinear conductance:
 eV   
ieV 

T dI / dV  sinh
 1  
2 2k BT 
 2 k BT  


 eV
 cot h
 2 k BT

 1
 
ieV 
 

Im  1  
2 2k BT 
 2

r.h.s. is only function of V/T
2
Evidence for Luttinger liquid
Yao et al., Nature 1999
Luttinger liquid properties
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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…
Fusion: Electron beam welding
(transmission electron microscope)
Fuhrer et al., Science 2000
Terrones et al., PRL 2002
Nanotube Y junctions
Li et al., Nature 1999
Landauer-Büttiker theory ?
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Standard scattering approach useless:
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Elementary excitations are fractionalized
quasiparticles, not electrons
No simple scattering of electrons, neither at
junction nor at contact to reservoirs
Generalization to Luttinger liquids
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Coupling to reservoirs via radiative boundary
conditions
Junction: Boundary condition plus impurities
Coupling to voltage reservoirs
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Two-terminal case,
applied voltage
eU   L   R
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Left/right reservoir injects `bare´ density of
R/L moving charges
 R0 ( L / 2)   L / 2vF
 L0 ( L / 2)   R / 2vF
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Screening: actual charge density is
 ( x)   R   L  g 2 (  R0   L0 )
Egger & Grabert, PRL 1997
Radiative boundary conditions
Egger & Grabert, PRB 1998
Safi, EPJB 1999
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Difference of R/L currents unaffected by
screening:  R ( x)   L ( x)   R0 ( x)   L0 ( x)
Solve for injected densities
boundary conditions for chiral density
near adiabatic contacts
 1

 1

eU
 2  1  R ( L / 2)   2  1  L ( L / 2)  
2vF
g

g

Radiative boundary conditions …
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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
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Landauer-Büttiker: Incoming and outgoing states
related via scattering matrix
out (0)  Sin (0)
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Difficult to handle for correlated systems
What to do ?
Some recent proposals …
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Perturbation theory in interactions
Lal, Rao & Sen, PRB 2002
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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)
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additional impurities generate arbitrary S matrices,
no conceptual problem Chen, Trauzettel & Egger, PRL 2002
Ideal symmetric junctions
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N>2 branches, junction with S matrix
 z 1 z

 z z 1
S 
...
...

 z
z


... z 

... z 
... ... 

... z  1
2
z
,  0
N  i
Crossover from full to no
transmission tuned by λ
implies wavefunction matching at node
1 (0)  2 (0)  ...  N (0)
j (0)  j ,in (0)  j ,out (0)
Boundary conditions at the node
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Wavefunction matching implies density
matching 1 (0)  ...   N (0)
can be handled for Luttinger liquid
Additional constraints:
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Kirchhoff node rule
Gauge invariance
I
i
0
i
Nonlinear conductance matrix
e I i
can then be computed exactly Gij 
h  j
for arbitrary parameters
Solution for Y junction with g=1/2
Nonlinear conductance:
8  Vi
2  V j


Gii  1 

1






U

U
i
j
9
9 j i 
with
eVi
 1 TB  ieU i  Vi / 2 
 Im  

2TB
2T
2

TB / D  w01/(1 g )
w0 ( N ,  ) 

2 N 2  2  2 N
N ( N  2)  2

Nonlinear conductance
g=1/2
1   F  eU
 2  3   F
Ideal junction: Fixed point
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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
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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
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Nonperturbative solution:
0
0
I1  I1  I , I 2,3  I 2,3  I / 2
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Asymmetry contribution


 1 WB  2i I10  I / 2 / e 

I  eWB Im   
2T
2

WB  W 2 / D
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Strong asymmetry limit:
I1  0, I 2,3  I
0
2, 3
I /2
0
1
Crossed tubes: Local Coulomb drag
Komnik & Egger, PRL 1998, EPJB 2001
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Different limit: Weakly coupled crossed
nanotubes
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Single-electron tunneling between tubes irrelevant
Electrostatic coupling relevant for strong
interactions, g  1 / 2
Without tunneling:
Local Coulomb drag
Hamiltonian for crossed tubes
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Without tunneling:
1
2
H  H Lutt
 H Lutt
 c cos 4g1 (0) cos 4g2 (0)


vF
2
2
H 
dx  i  (i )

2g
 Rotated boson fields:
i
Lutt
 

 ( x)  1 ( x)  2 ( x)/ 2
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
Boundary condition decouples:U  (U1  U2 ) / 2
Hamiltonian also decouples!
Map to decoupled 2-terminal models
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Two effective two-terminal (single impurity)
problems for g  2 g
H  H  H
H  H
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
Lutt

 c cos 8g (0)

Take over exact solution for two-terminal
problem
Dependence of current on cross voltage?
Crossed tubes: Conductance
G11
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 nanotubes
Kim et al., J. Phys. Soc. Jpn. 2001
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Measure nonlinear
conductance G11 for
cross voltage
 20meV  U 2  20meV
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Zero-bias anomaly for
small cross voltage
Conductance dip
becomes peak for
larger cross voltage
U1 (m eV)
Coulomb drag: Transconductance
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Strictly local coupling: Linear transconductance G21 always vanishes
Finite length: Couplings in +/- sectors differ
c
c  c 
dx cos2( k F ,1  k F , 2 ) x 

L L / 2
L/2
1 /(1 2 g )
c

T / D    
 D
TB  TB Now nonzero linear transconductance,

B
except at T=0!
Linear transconductance: g=1/4

B
T 1
1
1  d  ' ( d   1 / 2)
G21   
2  1  d  ' ( d   1 / 2)
d   TB / 2T
Absolute Coulomb drag
Averin & Nazarov, PRL 1998
Flensberg, PRL 1998
Komnik & Egger, PRL 1998, EPJB 2001
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For long contact & low temperature:
Transconductance approaches maximal
value

B

B
G21 (T  0, T / T  0)  1/ 2
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At zero temperature, linear drag conductance
vanishes (in not too long contact)
G21 (T  0)  0
Coulomb drag shot noise
Trauzettel, Egger & Grabert, PRL 2002
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Shot noise at T=0 gives important information
beyond conductance
P ( )   dte it I (t )I (0)
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For two-terminal setup, one weak impurity,
DC shot noise carries no information about
fractional charge P  2eI (U )
BS
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Crossed nanotubes: For U1  0,U 2  0  P1  0
must be due to cross voltage (drag noise)
Shot noise transmitted to other tube ?
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Mapping to decoupled two-terminal problems
implies
I (t )I (0)  0

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
Consequence: Perfect shot noise locking
P1  P2  ( P  P ) / 2
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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?
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Russian doll structure, electronic transport in
MWNTs usually in outermost shell only
Typically 10 transport bands due to doping
Inner shells can create `disorder´
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Experiments indicate mean free path   R...10 R
Ballistic behavior on energy scales
E  1
   / vF
MWNTs: Ballistic limit
Egger, PRL 1999
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Long-range tail of interaction unscreened
Luttinger liquid survives in ballistic limit, but
Luttinger exponents are closer to Fermi
liquid, e.g.   1 N
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
Bachtold et al., PRL 2001
(Basel group)
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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
Tunneling density of states: MWNT
Basel group, PRL 2001
Geometry dependence
end  2bulk
Interplay of disorder and interaction
Egger & Gogolin, PRL 2001, Chem. Phys. 2002
Rollbühler & Grabert, PRL 2001
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Coulomb interaction enhanced by disorder
Microscopic nonperturbative theory:
Interacting Nonlinear σ Model
Equivalent to Coulomb Blockade: spectral
density I(ω) of intrinsic electromagnetic

dt
modes
P( E )  Re 
0


J (T  0, t )  
0
expiEt  J t 
d



I ( ) e it  1
Intrinsic Coulomb blockade
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TDOS
Debye-Waller factor P(E):
 E / k BT
 ( E)
1 e
  dPE   
 / k BT
0
1 e
For constant spectral density: Power law with
exponent   I (  0)
Here:
1/ 2
2
U0


* n
*
I ( ) 
Re

i

/
D


D

D


2

*
R
2 ( D  D)

n 



D / D  1  0U 0 , D  v  / 2
*
2
F


Field/charge diffusion constant
Dirty MWNT
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High energies: E  EThouless  D /(2R)
Summation can be converted to integral,
yields constant spectral density, hence power
R
law TDOS with

ln D* / D
2 0 D
2

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
Tunneling into interacting diffusive 2D metal
Altshuler-Aronov law exponentiates into
power law. But: restricted to   R
Numerical solution
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Power law well below
Thouless scale
Smaller exponent for
weaker interactions,
only weak dependence
on mean free path
1D pseudogap at very
low energies
  10R,U 0 / 2vF  1, vF / R  1
Conclusions
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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