lectures 18-19 carbon nanotubes - lancaster university 18-19 carbon nanotubes and graphene physical...
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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
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.
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