1. NANOTUBI DI CARBONIO : struttura,
proprietà, sintesi, applicazioni…..
(SEMINARIO di CHIARA CASTIGLIONI)
Here we have what is almost certainly the strongest, stiffest, toughest
molecule that can ever be produced, the best possible molecular
conductor of both heat and electricity. In one sense the carbon
nanotube is a new man-made polymer to follow on from nylon,
polypropylene, Kevlar. In another, it is a new “graphitic” filler, but
now with the ultimate possible strength. In yet another, it is a new
species in organic chemistry, and potentially in molecular biology as
well, a carbon molecule with the almost alien property of electrical
conductivity, and super-steel strength.
R.E. Smalley, Chemistry Nobel 1996

2. Phase diagram of carbon emphasizing graphite, cubic diamond, and hexagonal diamond
phases, as well as liquid carbon. Solid lines represent equilibrium phase boundaries.
A: commercial synthesis of diamond from
graphite by catalysis;
B: P=T threshold of very fast (<1 ms) solid-
solid transformation of graphite to diamond;
C: P=T threshold of very fast transformation of
diamond to graphite;
D: single crystal hexagonal graphite transforms
to retrievable hexagonal-type diamond;
Pressure (GPa)
E: upper ends of shock compression/quench
cycles that convert hex-type graphite particles
to hex-type diamond;
F: upper ends of shock compression/quench
cycles that convert hex-type graphite to cubic-
type diamond;
B, F, G: threshold of fast P=T cycles, however
generated, that convert either graphite or
hexagonal diamond into cubic-type diamond;
H, I, J: path along which a single crystal hex-
type graphite compressed in the c-direction at
room temperature loses some graphite
characteristics and acquires properties
consistent with a diamond-like polytype, but
reverts to graphite upon release of pressure.

3. OTHER CARBON MATERIALS
fullerenes
APPLICATIONS
– fullerenes
electronics – nanotubes
– amorphous carbons
energy storage,
– carbon nanotubes
batteries,
nanotubes – porous graphites
sensors
Carbon nanotubes, M.S. Dresselhaus, G. Dresselhaus,
mechanical and – carbon fibers,
Ph. Avouris (Eds.) Springer (2001)
tribological amorphous carbons
and DLC hard coatings
applications
– micro and nano
crystalline graphites
D. Donadio, L. Colombo, P. Milani, G. Benedek,
– carbon fibers
Phys. Rev. Lett., 83, 776-779 (1999)
– glassy carbon
“graphitic”
– porous graphites
– carbon black
mixed – amorphous carbons
sp2, sp3, sp – diamond like
disordered carbons carbons (DLC)
C atoms

4. Legame σ tra orbitali atomici di tipo np (pz)

5. Legame π tra orbitali atomici di tipo np (px,py)

6. L'ibridazione nel carbonio
C (Z = 6)
configurazione elettronica: 1s2 2s2 2p2
1s2 shell K
alto potenziale di ionizzazione
non e' interessata alla
formazione del legame chimico
2s2 2p2 shell L
incompleta, a piu' alta energia
(minore potenziale di ionizzazione)
Responsabile del legame chimico

7. Ibrido sp3: lobi diretti nello spazio secondo i vertici di un
tetraedro il cui centro corrisponde al nucleo del carbonio
(2s + 2 px + 2 p y + 2 pz )
1
ψ1 =
2
ψ 2 = (2 s + 2 px − 2 p y − 2 pz )
1
2
ψ 3 = (2 s − 2 px + 2 p y − 2 pz )
1
2
ψ 4 = (2 s − 2 px − 2 p y + 2 pz )
1
2
Si ottengono 4 orbitali ibridi dalla combinazione di 1 orbitale s
con 3 orbitali p notazione sp3

8. Giustificazione dell’orientamento spaziale degli orbitali ibridi sp3
(2s + 2 px + 2 p y + 2 pz )
1
ψ1 =
z 2
ψ 2 = (2 s + 2 px − 2 p y − 2 pz )
1
(-1,-1,1)
2
4
ψ 3 = (2 s − 2 p x + 2 p y − 2 pz )
1
(1,1,1)
2
1
ψ 4 = (2 s − 2 px − 2 p y + 2 pz )
1
(0,0,0) 2
3
(-1,1,-1)
2
y
(1,-1,-1)
x

9. Ibrido sp2
3 elettroni di valenza
1 elettrone di valenza
( )
1
ψ1 = 2 s + 2 ⋅ 2 px 2 pz
3
1⎛ ⎞
1 3
⎜ 2s − ⋅ 2 py ⎟
ψ2 = ⋅ 2 px +
⎜ ⎟
2
3⎝ 2 ⎠
1⎛ ⎞
1 3
⎜ 2s − ⋅ 2 py ⎟
ψ3 = ⋅ 2 px −
⎜ ⎟
2
3⎝ 2 ⎠

10. Ibrido sp
1
(2 s + 2 pz )
ψ1 =
2
Restano non ibridizzati Ibrido sp
1
(2 s − 2 pz )
ψ2 =
2 px 2 p y 2

11. Esempio di molecole con carbonio in stato di
ibridazione sp3: metano

12. Esempio di molecole con carbonio in stato di
ibridazione sp3: etano

13. Esempi di ibridi sp2: copresenza di legame σ e π
Etilene,
σ
H2C=CH2
π

14. Esempi di ibridi sp2: copresenza di legame σ e π
σ
Butadiene,
H2C=(CH)-(CH)=CH2
π

15. Esempi di ibridi sp2: copresenza di legame σ e π
σ
Benzene,
C6H6
π

16. Esempi di ibridi sp: copresenza di legame σ e π
Acetilene,
H-C≡C-H
Triplo legame:
1 di tipo σ,
2 di tipo π

17. Struttura del diamante (ibridazione sp3)

18. ALCANI (ibridazione sp3)
butano (n=4)
metano (n=1)
etano (n=2)
pentano (n=5)
propano
(n=3)
esano (n=6)

19. Struttura della grafite (ibridazione sp2)
Asse c
Vista lungo l'asse c

20. POLYCONJUGATED MOLECULES
1D π conjugated systems: 2D π conjugated systems:
polyenes PAH, Polycyclic Aromatic Hydrocarbons
conjugated 2pz orbitals

21. Grafite nanostrutturata
PAH
Polycyclic
Aromatic
Hydrocarbons

22. Nanotubi di carbonio
(ibridazione prevalente sp2)

23. Stable forms of carbon clusters: (a) a piece of a graphene sheet,
(b) the fullerene C60, and (c) a model for a carbon nanotube.

24. Graphene ribbons terminated by (a) armchair edges and (b)
zigzag edges, indicated by filled circles. The indices denote the
atomic rows for each ribbon.

25. High-resolution electron
micrographs of graphitic particles
(a) as obtained from an electric
arc deposit, the particles display
a well-defined faceted structure
and a large inner hollow space
(b) the same particles after being
subjected to intense electron
irradiation. The particles now
show a spherical shape and a
much smaller central empty space.

26. Sketch of the cross section of a
PAN carbon fiber along the
fiber axis direction.
Here the in-plane (La) and c-
axis (Lc) structural coherence
lengths are indicated.

27. Schematic model for the microstructure of activated carbon fibers
Fiber after some heat
High surface area fiber
treatment, showing partial
where the basic structural
alignment of the basic
units are randomly
structural units.
arranged

28. Nanostructured amorphous carbon films
D. Donadio, L. Colombo, P. Milani, G. Benedek,
Phys. Rev. Lett., 83, 776-779 (1999)

29. Multi-walled carbon nanotubes
S.Ijima, Nature 358,
220 (1991)
Nanotubi cresciuti sul catodo
durante una scarica ad arco tra 2
elettrodi di grafite (T≈ 3000 K)
Reference book:
Carbon nanotubes, M.S. Dresselhaus, G. Dresselhaus,
Ph. Avouris (Eds.) Springer (2001)

30. Multi-walled carbon nanotubes

31. Fullerenes within SWNTs: peapods
La@C82

32. Heat treatment of peapods produces
double-wall NT

33. Carbon nanotubes,
M.S. Dresselhaus, G. Dresselhaus,
Ph. Avouris (Eds.) Springer (2001)

34. (5,5)
(9,0)
(10,5)

35. Ch = 4 a1+ 2 a2 nanotubo (4,2)

36. Rosso: 3,3 armchair, θ=45°

37. Rosso: (5,0) zig-zag θ=0°

38. The unrolled honeycomb lattice of an Armchair nanotube
structural unit
Ch = Chiral vector
T = Translation vector (k)

46. Electronic 1D density of states per unit cell
of a 2D graphene sheet for two (n,0) zigzag
nanotubes:
(a) the (10,0) nanotube which has
semiconducting behaviour,
(b) the (9,0) nanotube which has metallic
behaviour.
Also shown in the is the density of states for the
2D graphene sheet (dotted line).

47. Derivative of the current-voltage dI/dV
curves obtained by scanning tunnelling
spectroscopy on various isolated single-wall
carbon nanotubes with diameters near 1.4nm.
Nanotubes #1 - 4 are semiconducting and #5
- 7 are metallic.

48. nanotube (n,m) → (4,2)
Ch = 4 a1+ 2 a2

49. Hamiltoniano elettronico H = H(θ1,θ2) alla Hückel
(i.e. tight-binding ristretto a orbitali 2pz)
(0,-1) (1,0)
T
1 2 1 2
1 2
a1
Ch
1 2 1 2
τ1 ϕ2 ϕ1
τ2 a2
(-1,0) (0,1)
θi = k•ai

50. Curve di dispersione elettronica (4,2)
π∗
ξ = -π
Energia in unità di β
K1
K2
μ=2
EF
K K ξ=0
M
ξ=π
μ=0
π K K
Γ0 μ = N- 1
μ=0
Funzione del numero quantico μ = 0μ..=25
3
μ=1
μ=1
K K

51. Curve di dispersione elettronica (6,3)
Energia in unità di β
NT conduttore
Funzione del numero quantico μ = 0 .. 41

52. Curve di dispersione elettronica (17,8)
Energia in unità di β
Funzione del numero quantico μ = 0..325

53. Densità di stati elettronici
di due nanotubi chirali
metallici
Van Hove
singularities
EF
(14,5)
(11,8)
EF

54. Analytic expressions for the electronic energies have been
obtained with a symmetry treatment of Pz orbitals in the
frame of Hückel Theory
ε ε
θ/π θ/π
[ ]}
{
ε p (θ , ϕ ) = m 3 + 2 cos ϕ ± 2 (1 + cos ϑ )(1 + cos ϕ )1 2
Zigzag: (10,0) Ch ≅ 2.42 nm 12
(θ , ϕ ) = m{3 + 2 cos ϑ ± 2[(1 + cos ϑ )(1 + cos ϕ ) ]}
Armchair: (10,10) Ch ≅ 4.2 nm εp 12 12

55. (10,10) Ch ≅ 4.2 nm
(10,0) Ch ≅ 2.42 nm

56. Wave function, Van Hove peak at energy -0.95 Beta units
Tube axis

57. Energy dispersion and
density of states for
(9,0) zigzag nanotube
Density of states for (150,150)
armchair nanotube
(150,150) Ch=63 nm

67. Figure 5: TEM micrographs of seaweed-like carbon
objects produced at 6.5 GPa and 950°C.
Figure 4: TEM micrograph (a) at low magnification
and (b), (c) at high magnification of MWNT treated at
5.5 GPa and 950°C.

70. Raman spectra of graphite and amorphous carbon
D G
Crystalline graphite
G
1200
1580
1100
1000
Raman Intensity
900
800
Absorbance
700
Raman Intensity
600
500
400
300
200
100
2000 1800 1600 1400
Wavenumbers (cm-1)
Wavenumbers (cm-1)
Disordered graphite
G
900
1573
850
800
Raman Intensity
750
700
650
600
Wavenumbers (cm-1)
Absorbance
550
500
450
Annealed amorphous carbon
400
D
350
courtesy of A.C. Ferrari
300
1330
250
200
Dept. of Engineering
150
100
Cambridge (UK)
2000 1800 1600 1400
Wavenumbers (cm-1)
Wavenumbers (cm-1)

71. A.M. Rao, E. Richter, S. Bandow, B. Chase, P.C. Eklund, K. W.
Williams, M. Menon, K. R. Subbaswamy, A. Thess, R. E. Smalley,
G. Desselhaus, M.S. Dresselhaus, Science 275 (1997) 187
Spettri Raman Risonanti di un campione
di nanonotubi singola parete contenente
nanotubi di diversi diametri

73. Room temperature RBM spectra for bundles
of SWNTs produced by pulsed
laser vaporization using an Fe/Ni catalyst in a
carbon target. Spectra (a)-(d) are
collected at fixed laser excitation energy (1.17
eV; Nd:YAG) from samples grown at
T = 780, 860, 920 and 1000 °C, respectively.
Note that the spectral weight shifts to
smaller RBM frequencies with increasing
growth temperature (Tg) indicating that
diameter
increases with increasing Tg. The intensities
and frequencies of the RBM bands in
spectra (e)-(g) collected from the same sample
(Tg=1000°C) but with different laser
excitation energies (488nm; 514.5nm; 647
nm; 1064nm) are quite different,
demonstrating how different diameter tubes
are excited as the excitation energy changes.

74. Raman spectroscopy is used to characterize carbon nanotubes;
the G band brings important structural information
G- is associated to
Studying a metal/semiconductor junction in a
metallic tubes: why ?
nanotube using space-resolved Raman
G+
G+
G-
Taken from:
S.K. Doorn et al., PRL 94, 016802 (2005)

75. Carbon nanotubes:
extended π-conjugated systems
long range electronic and vibrational interactions
crucial dependence of the electronic structure on the
geometric structure (n,m)
phonons do experimentally depend on the diameter and
electronic structure of the tube
⇒ Fairly challenging system to model !

76. Polyconjugated carbon systems
Polyenes
Raman dispersion with chain length
Graphite & Carbon Nanotubes
Kohn Anomalies and Electron-Phonon Interaction in
Graphite (S. Piscanec, M. Lazzeri, F. Mauri, A. C. C. Castiglioni, et al. Phyl. Trans. R. Soc. Lond. A., 362
Ferrari, and J. Robertson, PRL, 93 (2004)) (2004)
… Polyynes also !
See poster 39-M,
M. Tommasini, A. Milani, A. Lucotti, M. Del Zoppo, C.
Castiglioni, G. Zerbi

77. Modeling electrons and phonons in carbon nanotubes
- Structural unit: 2 atoms A general treatment for
- Screw axis symmetry any carbon nanotube
- Real (curved) geometry (n,m)
Calculation of phonons on
Bloch theorem and
the basis of valence
nanotube boundary
coordinates
conditions
GFL = Lω2
(with curved geometry)
- band structure - phonon dispersion
- DOS - vibrational displacements
- phonon DOS

78. Describing the geometry of a generic (n,m) nanotube
Ch = 4 a1+ 2 a2 (4, 2) nanotube

79. (14,5)
Electronic band structure of
semiconducting (4,2) nanotube
π∗
ξ = -π
Energy (units of β)
K1
EF
K2
μ=2
EF
K K ξ=0
M
ξ=π
μ=0
π K K
Γ0 μ = N- 1
μ= 0
Function of the quantum numbersμμ,ξ
=3
μ=1
μ= 1
K K

80. Ohno’s three parameters force field (1) generalised to graphite (2)
(1) K. Ohno, J. Chem. Phys. 95, 5524 (1995)
(2) C. Mapelli, C. Castiglioni, G. Zerbi, K. Müllen, Phys. Rev. B (1999)
semiempirical parameters
bond stretching force constants
bond
∂ 2 Eπ
order
bond-bond
Π ij ≡
polarizability
∂β i ∂β j
{[c * (θ1 ,ϑ2 )ceσ (ϑ1 ' ,ϑ2 ' ) + c *0σ (θ1 ,ϑ2 )ceν (ϑ1 ' ,ϑ2 ' )][c0λ (θ1 , ϑ2 )c *eμ (ϑ1 ' , ϑ2 ' ) + c0 μ (θ1 , ϑ2 )c *eλ (ϑ1 ' ,ϑ2 ' )] + c.c.}
π π π π
1
(2π ) ∫π ∫π ∫π ∫π
0ν
dϑ1 dϑ2 dϑ1 ' dϑ2 '
Π λμ ,νσ =
ε 0 (θ1 , ϑ2 ) − ε e (ϑ1 ' , ϑ2 ' )
4
− − − −
electronic structure (Hückel)
The vibrational force field
is coupled to the
electronic structure

81. Phonon dispersion curves of graphite S. Piscanec, M. Lazzeri, F. Mauri, A. C.
Ferrari, and J. Robertson, PRL, 93 (2004)
Kohn anomaly and long range
interactions Kohn Anomalies and Electron-Phonon
Interaction in Graphite
∂ 2 Eπ Long range
Π ij ≡ stretching force
∂β i ∂β j constants
Ohno
force field;
variable
threshold on
fij

82. Generalization of the Ohno Force Field to nanotubes
of any diameter and chirality
Method based on graphene cell (2 atoms) + screw axis symmetry
The correct long range behavior of the force field is dictated by
the electronic-structure dependent bond-bond polarizabilities Π:
Brillouin zone integration
Boundary conditions:
Geometrical
parameters of
the (n,m) tube

83. Bond-bond polarizabilities Πij
∂ 2 Eπ It is directly related to stretching
Π ij ≡
∂β i ∂β j force constants
Metallic: slow decay
Semiconducting: fast decay

84. The G matrix is specific for any given nanotube:
G = G(n,m)
tube curvature
The F matrix is specific for any given nanotube
electronic structure (Πij):
F = F(n,m)

85. Raman spectra of individual
G band:
single wall nanotubes
different frequency dispersion law
metallic
(while changing the tube diameter)
observed for metallic and
(18,9)
semiconducting nanotubes
G+ longitudinal ?
(diameter
independent…)
(19,1)
semiconducting
(11,2) G- transversal ?
(17,7)
(dramatically
diameter
dependent…)
(17,3)
(15,2)
All data shown are taken from:
A. Jorio, A. G. Souza Filho, et al.,
Phys. Rev. B, 65, 155412 (2002)

86. Experimental findings Empirical force field
by Jorio et al. (armchair tubes)
A. Jorio, et al., Phys. Rev. B, 65, 155412 (2002)
independent theoretical works
by M. Lazzeri et al. PRB 73,
transversal
155426 (2006)
longitudinal
Large longitudinal/transversal
splitting: favourably compares
with experiments and…
longitudinal G- transversal G+

87. Dispersion of the G line Full symbols: longitudinal phonons
Open symbols: transversal phonons
Cold colours: metallic CNTs
Warm colours: semiconducting CNTs
μ=0
μ=1

88. Phonons of the chiral (6,3) metallic tube

89. Conclusions
1. Carbon nanotubes share long range interaction
physics similarly to other π-conjugated systems
(polyacetylene, graphite)
2. A successful and general model of phonons in
nanotubes has been introduced which couples
to the electronic structure of the given (n,m) tube
3. The correct longitudinal/transversal splitting of the G
phonon as a function of tube diameter is found.
The assignment of the long./transv. character of G
phonons for general tubes is proposed