An introduction to the theory of Carbon nanotubes

A. De Martino

Institut für Theoretische Physik

Heinrich-Heine Universität

Düsseldorf, Germany

Overview

Introduction

Band structure and electronic properties

Low-energy effective theory for SWNTs

Luttinger liquid physics in SWNTs

Experimental evidence

Summary and conclusions

Allotropic forms of Carbon

Curl, Kroto, Smalley 1985

Iijima 1991graphene

(From R. Smalley´s web image gallery)

Classification of CNs:single layer

Single-wall Carbon nanotubes (SWNTs,1993)- one graphite sheet seamlessly wrapped-up to form a cylinder

- typical radius 1nm, length up to mm

(From R. Smalley´s web image gallery)

(From Dresselhaus et al., Physics World 1998)

(10,10) tube

Classification of CNs: ropes

Ropes: bundles of SWNTs- triangular array of individual SWNTs

- ten to several hundreds tubes

- typically, in a rope tubes of different diameters and chiralities

(From R. Smalley´s web image gallery)(From Delaney et al., Science 1998)

Classification of CNs: many layers

Multiwall nanotubes (Iijima 1991)

- russian doll structure, several inner shells

- typical radius of outermost shell > 10 nm

(From Iijima, Nature 1991) (Copyright: A. Rochefort, Nano-CERCA, Univ. Montreal)

Why Carbon nanotubes so interesting ?

Technological applications- conductive and high-strength composites

- energy storage and conversion devices

- sensors, field emission displays

- nanometer-sized molecular electronic devices

Basic research: most phenomena of mesoscopic physics observed in CNs- ballistic, diffusive and localized regimes in transport

- disorder-related effects in MWNTs

- strong interaction effects in SWNTs: Luttinger liquid

- Coulomb blockade and Kondo physics

- spin transport

- superconductivity

Band structure of graphene

Tight-binding model on hexagonal lattice

- two atoms in unit cell

- hexagonal Brillouin zone

(Wallace PR 1947)

iRRBRA

cHcctHi

,,,0 ..

nm 0.142d ; nm 246.0a

eV37.2

t1

23

K K

Band structure of graphene

Tight-binding model- valence and conduction bands touch at E=0

- at half-filling Fermi energy is zero (particle-hole symmetry): no Fermi surface, six isolated points, only two inequivalent

Near Fermi points

relativistic dispersion relation

Graphene: zero gap semiconductor

m/s108,

,5

FF

F

vKkq

qvqE

(Wallace PR 1947)

Structure of SWNTs: folding graphene

(n,m) nanotube specified by wrapping, i.e. superlattice vector:

Tube axis direction

21 amanC

2211 aatT

)2,2gcd(

/)2(

/)2(

2

1

nmmnd

dmnt

dnmt

R

R

R

Structure of SWNTs

(n,n)

armchair

(n,0)

zig-zag

chiral

(4,0)

Electronic structure of SWNTs

Periodic boundary conditions → quantization of

- nanotube metallic if Fermi points allowed wave vectors,

otherwise semiconducting !

- necessary condition: (2n+m)/3 = integer

k

armchair zigzag

metallic semiconducting

Electronic structure of SWNTs

Band structure predicts three types:

- semiconductor if (2n+m)/3 not integer; band gap:

- metal if n=m (armchair nanotubes)

- small-gap semiconductor otherwise (curvature-induced gap)

Experimentally observed: STM map plus conductance measurement on same SWNT

In practice intrinsic doping, Fermi energy typically 0.2 to 0.5 eV

eV13

2

R

vE F

Density of states

Metallic tube:

- constant DoS around E=0

- van Hove singularities at opening of new subbands

Semiconducting tube:

- gap around E=0

Energy scale ~1 eV

- effective field theories valid for all relevant temperatures

Metallic SWNTs: 1D dispersion relation

Only subband with relevant, all others more than 1 eV away

- two degenerate Bloch waves, one

for each Fermi point α=+/-

- two sublattices p =+/-, equivalent

to right/left movers r =+/-

- electron states

- typically doped:

0k

r

KkvEq FFFF

/

SWNTs as ideal quantum wires

Only one subband contributes to transport

→ two spin-degenerate channels

Long mean free paths

→ ballistic transport in not too long tubes

No Peierls instability

SWNTs remain conducting at very low temperatures → model systems to study correlations in 1D metals

m1

Conductance of ballistic SWNTs

Landauer formula: for good contact to voltage reservoirs, conductance is

Experimentally (almost) reached- clear signature of ballistic transport

What about interactions?

h

e

h

eNG bands

22 42

Including electron-electron interactions in 1D

In 1D metals dramatic effect of electron-electron interactions: breakdown of Landau´s Fermi liquid theory

New universality class: Tomonaga-Luttinger liquid

- Landau quasiparticles unstable excitations

- stable excitations: bosonic collective charge and spin density fluctuations

- power-law behaviour of correlations with interaction dependent exponents → suppression of tunneling DoS

- spin-charge separation; fractional charge and statistics

- exactly solvable by bosonization

Experimental realizations: semiconductor quantum wires, FQHE edges states, long organic chain molecules, nanotubes, ...

(Tomonaga 1950, Luttinger 1963, Haldane 1981)

Luttinger liquid properties

1d electron system with dominant long-range interaction

- charge and spin plasmon densities; - spin-charge separation !

- depends on interaction :

Suppression of tunnelling density of states at Fermi surface :

- exponent depends on geometry (bulk or edge)

ExtxedtEx iEt

)0,(),(Re1

),(0

22222LL 22 sxs

Fcxcc

F dxv

Kdxv

H

sxcx ,

cK1

1

c

c

K

K no interaction

repulsive interaction

Field theory of SWNTs

Low-energy expansion of electron field operator:

- Two degenerate Bloch states

at each Fermi point

Keep only the two bands at Fermi energy,

Inserting expansion in free Hamiltonian gives

)()()( ''0 xxdxivH pppxxpF

1

0;

0

1

2,

BA

p

rKi

pR

eyx

p

pp yxFyxyx ),(,,

(Egger and Gogolin PRL1997, Kane et al. PRL 1997)

0k

)(),( xyxF pp

Coulomb interaction

Second-quantized interaction part:

Unscreened potential on tube surface

2222

2

2

´sin4´)(

/

zaR

yyRxx

eU

rrrrUrrrdrdH I

´´´´

2

1´´

´

1D fermion interactions

Momentum conservation allows only two type of processes away from half-filling

- Forward scattering: electrons remain at same Fermi point, probes long-range part of interaction

- Backscattering: electrons scattered to opposite Fermi point, probes short-range properties of interaction

- Backscattering couplings scale as 1/R, sizeable only for ultrathin tubes

Backscattering couplings

Fk2

Momentum exchange

Fq2

Coupling constant

Reaf /05.0/ 2Reab /1.0/ 2

Bosonization:

four bosonic fields (linear combinations of )- charge (c) and spin (s)

- symmetric/antisymmetric K point combinations

Bosonized Hamiltonian

Effective low-energy Hamiltonian

termsnonlinear2

222 aaaa

F Kv

dxH

),,,( ssccaa

(Egger and Gogolin PRL1997, Kane et al. PRL 1997)

rFF ixirqxki

r e 4~

r

Luttinger parameters for SWNTs

Bosonization gives

Long-range part of interaction only affects total charge mode

- logarithmic divergence for unscreened interaction, cut off by tube length

very strong correlations !

3.02.02ln8

12/12

RL

v

eKg

Fc

1caK

Phase diagram (quasi long range order)

Effective field theory can be solved in practically exact way

Low temperature phases matter only for ultrathin tubes or in sub-mKelvin regime

bF RRbvbB

bf

eDeTk

TbfT//

)/(

Theoretical predictions

Suppression of tunneling DoS:

- geometry dependent exponent:

Linear conductance across an impurity:

Universal scaling of scaled non-linear conductance across an impurity as function of

EEx ),(

4/)1/1(

8/2/1

g

gg

end

bulk

endTTG 2)(

TkeV B/

Evidence for Luttinger liquid

(Yao et al., Nature 1999)

gives 22.0g

end2

bulk

Conclusions

Effective field theory + bosonization for low-energy properties of SWNTs

Very low-temperature : strong-coupling phases

High-temperature : Luttinger liquid physics

Clear experimental evidence from tunnelling conductance experiments