electronic transport in one-dimensional wires

Electronic transportin one-dimensional wires

Akira Furusaki (RIKEN)

Aug 14, 2003 Electronic transport in 1D wires 2

Outline Tomonaga-Luttinger (TL) liquid Bosonization Single impurity in a TL liquid Two impurities in a TL liquid

linear conductance G

Random-matrix approach to transport in disordered wires


1D metals= Tomanaga-Luttinger liquid No single-particle excitations Collective bosonic excitations

spin-charge separation charge density fluctuations spin density fluctuations

Power-law decay of correlation functions (T=0)

tunneling density of states


TL liquids are realized in: Very narrow (single-channel) quantum wires edge states of fractional quantum Hall liquids Carbon nanotubes


Interacting spinless fermions Simplified continuum model

　　　　　　　　　　　　　　　　　　　　　　　　　　　　 kinetic energy

short-range repulsive interaction (forward scattering)


Abelian Bosonization Fermions = Bosons in 1D


Electron density


Kinetic energy


Bosonized Hamiltonian

TL liquid parameter g

g < 1: repulsive interaction FQHE edgeg = 1: non-interacting case g > 1: attractive interaction

Interacting fermions = free bosons

12

1

m


Correlation functions 　（ T=0 ）

Scaling dimension of is

4a


Single impurity

Non-interacting case (free spinless fermions)

transmission probability


Current

Conductance G changes continuously. no temperature dependence. is a marginal perturbation


Interacting spinless fermions reflection at the barrier potential

Hamiltonian

free boson + = pinning of charge density wave

electric current


Partition function (path integral)

effective action for

linear: dissipation due to gapless excitations in TL liquid

(Caldeira-Leggett: Macroscopic Quantum Coherence)

a particle (with coordinate ) moving in a cosine potential with friction

0x

|| n


Renormalization-group analysis Weak-potential limit weak perturbation:

scaling equation (lowest order): renormalized potential:

conductance

4cos0V


Strong-potential limit (weak-tunneling limit) duality transformation [A. Schmid (’83); compact QED by A.M. Polyakov]

　　　“ dilute instanton (=tunneling) gas”

t: tunneling matrix element (fugacity)


scaling equation:

renormalized tunneling matrix element:

conductance


Flow diagram for transmission probability (Kane & Fisher, 1992)

g<1 (repulsive int.) perfect reflection at T=0

g=1 (free fermions) marginal

g>1 (attractive int.) perfect transmission at T=0

1

01

Trans.Prob.

g


Exact results “Toulouse limit” g=1/2

introduce new fields

refermionization

quadratic Hamiltonian

cf. 2-channel Kondo problem (Emery-Kivelson, 1992)


Conductance at g=1/2

General gThe boundary sine-Gordon theory is exactly solvable (Ghoshal & Zamolodchikov, 1994)

Bethe ansatz elastic single-quasiparticle S-matrix (Fendley, Ludwig & Saleur, 1995)


Spinful case (electrons)(Furusaki & Nagaosa, 1993; Kane & Fisher, 1992)

charge boson: spin boson:

Hamiltonian

: non-interacting electrons

: repulsive interactions

: if spin sector has SU(2) symmetry


Weak-potential limit

Strong-potential limit (weak-tunneling limit)single-electron tunneling: t

RG flow diagram

critical surface

at intermediate

coupling

tKKdl

dt

11

2

11

00

2

11 VKK

dl

dV

1K 21 K

211

2

KKTtG

0

1

0

1

1

Trans.Prob.

Trans.Prob.

K K


External leads (Fermi-liquid reservoir) (Maslov & Stone, 1994)Tomonaga-Luttinger liquid:

Fermi-liquid leads:

Action

Current I vs Electric field E

dc conductance is not renormalized by the e-e interaction

if the wire is connected to Fermi-liquid reservoirs

Lx 0Lxx ,0


Weak e-e interactions (Matveev, Yue & Glazman, 1993)

small parameter:

V(q): Fourier transform of interaction potential

scaling equation for the transmission probability

lowest order in

but exact in

conductance


Coulomb interactions (Nagaosa & Furusaki, 1994; Fabrizio, Gogolin & Scheidel, 1994) : width of a quantum wire

scaling equation for tunneling

conductance

stronger suppression than power law

W

)/1log()( qWqV

|)|/log(11nF WvrK

trldl

dt 11

2

1

Fv

er

2

2/32/1 log

3

2exp

WT

vrG F


Experiments on tunneling Edge states in FQHE

(Chang, Pfeiffer & West, 1996)

tunneling between a Fermi liquid and edge state

　　　 [Fig. 1 & Fig. 2 of PRL 77, 2538 (1996) were shown in the lecture]

3/1

1

VI


Single-wall carbon nanotubes Yao, Postma, Balents & Dekker, Nature 402, 273 (1999)

[Fig. 1 and Fig. 3 were shown in the lecture.]

Segment I & II: bulk tunneling

Across the kink: end-to-end tunneling

exp:

TG


Resonant Tunneling (Double barriers) Non-interacting case

transmission amplitude: t

has maximum when resonance

(symmetric barrier)

symmetric case backscattering is irrelevant

asymmetric case backscattering is marginal = single impurity

0 dx

L R


When

life time of discrete levels

Conductance

if coherent tunneling

if

incoherent

sequential tunneling

peak width

1||,|| 22 RL tt

22

2

41

||

RL

RLt

2,, ||

2 RLF

RL td

v

22

2

41

RL

RL

EdE

dfdE

h

eG

RLT ,

TRL ,

22

2

41

RL

RL

h

eG

d

df

h

eG

RL

RL )(22

1

max

111

RLT

G

T

RL 2

1


Resonant tunneling in TL liquidsSpinless fermions

Hamiltonian

gate voltage

Current

Excess charge in [0, d ]

is massive

dt

deI

02

1 d 0~ d

~e

Q

~


Weak-potential limit (Kane & Fisher, 1992)

effective action for

single-barrier problem

scaling equation

if (symmetric) and (on resonance)

dvn /||

,01 V RL VV 1)cos(

g g1 1/4

1V 2V


Resonance line shape symmetric

¼ < g < 1 is the only relevant operator, on resonance

universal line shape peak width

not Lorentzian

RL VV


Weak-tunneling limit (Furusaki & Nagaosa, 1993; Furusaki,1998)

Off resonance

process is not allowed at low T

virtual tunneling

On resonancesequential tunneling

life time due to tunneling through a barrier

peak width

1t

e

2t

21

2

tt

TG

on :

11

21

gTt


Phase diagram at T=0 Symmetric barriers

Asymmetric barriers

g<1 g=1 g>1

G

g0

1

11/21/4

Transmissionprobability

GG

G

h

e2

11 )1( Vgdl

dV

22 1

1 tgdl

dt

h

e2G G G

0 1

1


T > 0 Weak potential

Weak tunneling

sequential tunneling


Experiments on resonant tunneling in TL liquids

Auslaender et al., Phys. Rev. Lett. 84, 1764 (2000)


Carbon nanotubes

Postma et al., Science 293, 76 (2001)


Summary In 1D e-e interaction is crucial

Tomonaga-Luttinger liquid Repulsive e-e interaction

backward potential scattering is relevant power-law suppression of tunnel density of states

Problems nontrivial fixed points at intermediate coupling Resonant-tunneling experiment? 　　

electronic transport in one-dimensional wires

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