One Dimensional Electronic Systems:

• Quantum Hall Edge States: Chiral Luttinger Liquid

• Quantum Wires• Semiconductor wires• Polyacetalene• Carbon Nanotubes

Integer Quantum Hall Effect with edges (Halperin 1982)

2x By k∼

2x By k∼

E

x

yν=1

EF

Bulk Landau levels:RL

( )( ) 1 2n x cE k nω= +

c

B

eB mc

c eB

ω =

=

Chiral Fermions ψR

( ) †1 v ; ; 4 2 2

Ri x Rx R t R x R R

c

L dx e nx

φ φκφ φ φ ψπ π π

∂= − ∂ ∂ + ∂ =∫ ∼

( ) ( )2 20Interactions: 2 only affect velocity v.x RV φ π∂

Fermi Liquid exponents (g=1) even with interactions

Fractional Quantum Hall Effect: Chiral Luttinger LiquidWen 1990

v

Incompressible Fluidν=1/m

Assume a single edge mode:

• Laughlin States ν=1/m• “Sharp” Edge

( )1 v 4 x R t R x RL dx

gφ φ φ

π= − ∂ ∂ + ∂∫

g is determinied by the Chiral Anomaly

Fractional Quantum Hall Effect: Chiral Luttinger LiquidWen 1990

Incompressible Fluidν=1/m

v

Single Mode approximation:Valid for• Laughlin States ν=1/m• “Sharp” Edge

J=σxyE

xE V= −∂

( )1 v +V(x)4 2

x Rx R t R x RL dx

gφφ φ φ

π π∂

= − ∂ ∂ + ∂∫Equation of motion :

2

v2 2 2x R x R

t x x

t x xy

g V

ej g E Eh

φ φπ π π

ρ σ

∂ ∂∂ + ∂ = − ∂

∂ + ∂ = =

1gm

ν= =

Fractional Quantum Hall Effect: Chiral Luttinger LiquidWen 1990

v

Incompressible Fluidν=1/m

Assume a single edge mode:

• Laughlin States ν=1/m• “Sharp” Edge

( )v 4 x R t R x RmL dx φ φ φπ

= − ∂ ∂ + ∂∫

( ) 1, ( ') ( ')2

x x x x xm

φ φ δπ

∂⎡ ⎤ = −⎢ ⎥⎣ ⎦

Electron op. (charge e)† ~ ime e φψ

Laughlin Q.P. op. (charge e/m)† ~ iQP e φψ

Quantum Point Contact: Analog of single barrier in L.L.

e

ν=1/m

L

R

Weak (electron) Tunneling( )† ~ R Lim

T R LH t te φ φψ ψ −=2 2 2

2 4

( ) ~~ for =1/3

mG T t Tt T ν

−

Weak (quasiparticle) Backscattering

quasiparticlee*=e/m

ν=1/m

( )†. ., . .,v ~ B Ti

T Q P B Q P TH te φ φψ ψ −=2 2/ 2

2 4 /3

( ) ~ v~ v for =1/3

mG T TT ν

−

−

T

B

Early point contact experiments (Milliken, Umbach and Webb 1994)

For weak tunnelingtheory predicts

G ~ T4

Resonance Lineshapes (Milliken, Umbach and Webb 1994)

( )2

2/3( , )3eG T G Th

δ δ=

Difficulty for interpretation: Gates produce a smooth confining potential

For “perfect” resonancestheory predicts:

P

GG

2/3Tδ

Cleaved Edge Experiments Chang, Pfeiffer and West, 1996

Tunneling from metal to ν=1/3 at an atomically sharp edge

V(µV) cV/T

Predicted Scaling Form:

( , ) ( / ); 3I V T cT F V Tαα α= =

c I(V

,T)/G

(T)

33 ( )F x x x= +

Fit from Experiment:

α ~ 2.7

Dependence of Tunneling Exponent on Filling Factor

Theory: Kane, Fisher 95Shytov, Levitov, Halperin 98

Exponent should depend on “neutral modes” predicted to be present forν≠1/m. Leads to plateau structurein α(ν)

Experiment: Grayson, et al, 1998

Exponent seems to depend only on the“charge mode”: α ~ 1/ν.

Experiments must not be in the low energy, long wavelengthlimit described by theory.

Shot Noise in a point contact: DePicciotto et. al.; Saminadayer et. al. 97

Direct measurement of the charge of the Laughlin Quasiparticle

ItStrong Pinch Offe tunneling

e-e

S = 2e It

Ib-e*

e*Weak Pinch OffQ.P. tunneling

S = 2e* Ib

e*=e/3

Quantum Wires:1D for E < ∆E: the subband spacing

1. Split Gate Devices: (Tarucha et al. 95) 2

2~2 *

Em L

∆

∆E ~ 5 meV

2. Cleaved Edge Overgrowth (Yacoby et al. 96)

∆E ~ 20 meV

Molecular Wire: Polyacetalene(See Heeger et al. Rev. Mod. Phys. 1988)

1E

k

E

k

EFEF ∆

Truly 1D .... but Peierl’s Instability:

Any commensurate 1D metal is unstable

Peirels Energy Gap:Elastic energy cost : ~ ∆2

Electronic energy gain: ~ λ∆2 log ω0/∆ ∆ ~ ω0 e-1/λ ~ 1.8 eV

“Armchair” [N,N] Carbon Nanotubes

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

3

E

k

• One Dimensional for

• Peierls Instability suppressed for large N ~ R/a:

Fv / ~ 1 eVE R<

Elastic energy cost : ~ N ∆2

Electronic energy gain: ~ 1 λ∆2 log ω0/∆∆ ~ ω0 e-N/λ

Low Energy Theory

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

3

RL

4 Channels:

{ }† †F

, ' ,v asR x asR asL x asL

a K K sH i ψ ψ ψ ψ

= =↑ ↓

= − ∂ − ∂∑ ∑

( ) ( )2 2

, ,,

1 1 v2 v t a s F x a s

a s F

L θ θπ

⎧ ⎫= ∂ − ∂⎨ ⎬

⎩ ⎭∑

† †Vαβγδ α β γ δψ ψ ψ ψElectron Interactions

1. “Backscattering Interactions”

• Violate Chiral symmetry• Have form λ cos(φα+φβ−φχ−φδ)• Can lead to electronic Instabilities

• Depend on the short range part of V(r) ( q ~ 1/a )• Suppressed by 1/N

2. “Forward Scattering Interactions”

• Preserve Chiral symmetry• Have form • Lead to Luttinger Liquid

• Depend on Long range part of V(r)• NOT suppressed by 1/N

Screened Coulomb Interaction :

x xn nα β α βφ φ∝ ∂ ∂

( )2 2int log /s totH e R R n=

Luttinger Liquid Theory

, ; ' 'K K K Kρ σθ θ θ θ θ± ↑ ↓ ↑ ↓= ± ± ± ±Charge/Spin/Flavor variables:

( ) ( )2 2

,

1 1 v2 v t x

a sL

gρ ρ ρ ρρ

θ θπ+ + +

⎧ ⎫⎪ ⎪= ∂ − ∂⎨ ⎬⎪ ⎪⎩ ⎭

∑1 Charge

Mode

( ) ( )2 2

,

1 1 v2 va t a F x a

a s F

L θ θπ

⎧ ⎫= ∂ − ∂⎨ ⎬

⎩ ⎭∑3 Neutral

Modes

12 281 log ~ .2

vs

F

RegRπ

−⎛ ⎞

= +⎜ ⎟⎝ ⎠

v v /F gρ =

Spin Charge Separation:ω

vFq

vρq

q

Tunneling Experiments

Metal contact to rope of nanotubes G(T) ~ TαBockrath et al. 1999

Theory:

αbulk= (g+1/g-2)/8 ~.25

αend= (1/g-1)/4 ~ .65

End contactedBulk contacted

0.3bulkα = 0.6endα =

Tunneling across a barrier in a single nanotube Yao et al. 99

Scaling Plot TαF(V/T)