Lectures 18-19

Carbon nanotubes

and graphene

Physical Properties of Carbon NanotubesSaitoh, Dresselhaus, Dresselhaus, Imperial College Press 1998

Layered poorly conducting semimetal used in pencils, fuses and nuclear fusion

moderators

Graphite

Graphene

Physical Properties of Carbon NanotubesSaitoh, Dresselhau, Dresselhaus, Imperial College Press 1998

NanotubesIjima 1991

Smalley 1993

σ - bonds

hybridisation forms strong directed bonds which determine a honeycomb lattice structure.

2sp

C

Carbon has 4 electrons in the outer s-p shell

ε

σ

*σ

strong covalent bonds

?

Ultra-thin graphitic films: from flakes to micro-devices

for references, see the review articleA.Geim & K.Novoselov - Nature Materials 6, 183 (2007)

Novoselov & Geim (Manchester)Science 306, 666 (2004)

σ - bonds

hybridisation forms strong directed bonds which determine a honeycomb lattice structure.

2sp

C

Carbon has 4 electrons in the outer s-p shell

)(πzp orbitals determine conduction properties of graphite

0γ

eV10~

ε

σ

*σ

pz-bands

Gr

'Gr

Valley

22yx

cond ppvvp +==ε

xpyp

Graphene (monolayer of graphite) is an atomically thin zero-gap

two-dimensional semiconductor with linear dispersion of

conduction and valence band electrons.

22yx

val ppvvp +−=−=ε

Electronic dispersion in the vicinity of the corner of the Brillouin zone: the same in both valleys.

Angle-resolved photo-emission spectroscopy of heavily doped graphene synthesized on silicon carbide

A. Bostwick et al – Nature Physics, 3, 36 (2007)

high-energy photon ħω~100-1000eV

Simultaneous detection of the energy, E and propagation angle θ of photo-electrons enablesone

to restorecompletely

the band structure.EAp

mEp=−+

=

)(cos2

||

||

εω

θ

h

work function

DoS

gatecarriers Vn ∝

holes electrons

Graphene-based field-effect transistor: GraFET

Geim and Novoselov, Nature Mat. 6, 183 (2007)

Wallace, Phys. Rev. 71, 622 (1947)

Graphene: gapless semiconductor

Filling factor

eBhc e

LL

e ρρρ

=xpyp

eBcr

nvn

cB

Bn

h

h

=≡

=±=±

)0(

...3,2,1,02

λ

λε

‘relativistic-type’Landau level spectrum

vpcond =ε

sec/10~ 8cmvvpvalence −=ε

-2-4 40

n (1012 cm-2)

2

2

4

6

0

ρ xx

(kΩ

)

graphene

b

E

p

σ xy

(4e2

/h) 1

2

-1

-2

-4

0

-3

4 a3

Quantum Hall effect in graphene-based

field-effect transistor.

Science 315, 1379 (2005)

Carbon nanotubes Iijima 1991Smalley 1993

STM images of carbon nanotubesT.W. Odom, J.-L. Huang, P.Kim, C.Lieber, Nature 391 (1998)

22yx

cond ppv +=ε

xpyp

22yx

val ppv +−=ε

Metallic nanotubes

nL

p

e

Lyy

y

LyiL

⊥

⊥

=

===⊥

⊥

hπ

ψ

ψψπ

2

~

)()0(/21

2

222

⊥

+=Lnhpv x

condnε

2

222

⊥

+−=Lnhpv x

valnε

,....3,2,1,0=n0=n

1=n

Fε

ε

xp

perimeter, 2πr

0=n

1=nε

xp

Metallic nanotubes – truly 1D conductors

Yao et al 1999

ε

Density of states

vhπγ 1

0 =

0=n

1=n 1=n|| xpv±

Semicondutor-type nanotubes

Depending on how the carbon sheet is rolled into a nanotube, the resulting nanotube may have a gap in

the electron spectrum. A gap in the nanotubespectrum is determined by its radius, which offers a

direct root towards engineering semiconductor wires with a prescribed band gap, for use in

electronic and optoelectronic devices.

ε

xp

tunnelling currentT.W. Odom, J.-L. Huang, P. Kim, C. Lieber, Nature 391 (1998)

gap

rvh

Promising applications of carbon nanotubes:

In surface tunnelling microscopy – used as a tip.

Make excellent tips for field-effect electron emitters (SONY).This uses the fact that electric field is always the highest near the end of a tip, and nanotubes make excellent electron guns for plasma displays.

equi-potential lines