QUANTUM COHERENT CONDUCTION IN CNTs

An Amateur’s View

T.Williams: SZFKI 19-09-2005

Learning Seminar Series on Carbon Nanotubes

CONTENTS

1. Four striking experiments on 1-D conduction in CNTs

2. Band structure of graphene and CNTs

3. Why armchair (n,n) CNTs are metallic

4. Quantum of conduction e2/h per 1-D channel

5. Ballistic 1-D thermal conduction and quantum of conductance (π2/3)kB2T/h

6. Evidence for superconductivity in CNTs

7. Indications of Tomonaga-Luttinger liquid behaviour in CNTs

1-D Quantum conductance and interference 1-D ballistic thermal conductance

1-D Superconductivity 1-D Tomonaga-Luttinger Liquid

Yu et al, Nano Letters 2005

Liang et al Nature 2001

Kociak et al PRL 2001Bockrath et al Nature 1999

Ropes

EF

GRAPHENE BAND STRUCTURE

Carbon: atom = 1s22s22p2 ; graphene = 1s2 (2s2p2)σ2pπ

2s2p2 σ bonding

2s + px

+ =+

+- +-

2s px 2s2p2 σx orbital

Carbon: atom = 1s22s22p2 ; graphene = 1s2 (2s2p2)σ2pπ

+-

+

-

2p π bonding

+

-

+

-

2p π* antibonding

2p π bonding

(n,n) tube: c=na1+na2

a1

a2

W-S Zone

k

k=3π/2a0

Armchair tubes are quasi-metallic

EF

k

Brillouin Zone

k1

K

M

k2

k

Γ

k

K point=Fermi point

kFP=kK=3π/2a0

E(k) around Fermi point

k

k

CARRIER DENSITY

Neutral system = quasi-metallic = zero gap semiconductor

4 n=2 electrons per atom X 2 atoms per unit cell = 8 electrons= 2 spin states X 4 bands filled to Fermi point

kK.ž = 3π/2a0

k

k

k

Degenerate semiconductor Electron metal Hole metal

EFermi

δQ = 0 δQ = +εe/unit cell δQ = -εe/unit cell

IMPOSING CHARGE

δQTotal = C Vg :

Vgate s

L

2R

C L/ln(R/s)

)/ln(20 sRRV

Se g

(S0=area of graphene unit cell)

If Efermi << Eexc subband , only four 1-D conduction channels = 2 bands X

QUANTUM OF CONDUCTANCE

1zN eV

Wk

kN

zN

zN

12

VveWk

kN

zev

zNev

zNIII 2

212

121

1221

Rsvr1EF+δeV

Rsvr2EF

I12

I21

No back scattering

eVEF

a b ba

21

2,

L

LkN

zand

kW

vbut ba

VheI

2

(Landauer formula)

Key: v = W/ħk

e2/h = gQ ≈ 40μS = 1/25kΩ

CONFIGURATION OF TRANSPORT EXPERIMENTS

VG

Al2O3

IDS

VDS

Au

Au, Pt

CNT Contacts: sometimes ohmic R kΩ GQ , quantum int. ,superc. often tunnel R 102 kΩ CB, TLL STM tunnel

QUANTUM OF CONDUCTANCE, QUANTUM INTERFERENCE

s1 s2

e-ikz

e+ikz

k = kK + (k/W)eVG = kK + eVG/ħvF

trrt

S

1-D THERMAL CONDUCTANCE

kvph

ω

k

Rsvr1T+δT

Rsvr2Temp T

I12

I21

No back scattering

Power flow per mode: TkLv

p Bph

Number of modes: dk

kN

Total energy flux:

ThTkdk

kNTk

Lv

pdNQ BT

Bph

21

ThTkQ B 22

3 Quantum of thermal conductance

per channel

At T<6K, 4 channels, at 100K estimate of 7 channels

1-D BALLISTIC PHONON CONDUCTION

EVIDENCE FOR SUPERCONDUCTIVITY

Nota: CNT ropes

EVIDENCE FOR TOMANAGA-LUTTINGER LIQUID BEHAVIOUR

Nota: CNT ropes

WHAT TO BELIEVE?

WHAT TO DO?

WHAT CAN WE DO?