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