NANOTUBI DI CARBONIO : struttura,
                        proprietà, sintesi, applicazioni…..
                        (SEM...
Phase diagram of carbon emphasizing graphite, cubic diamond, and hexagonal diamond
                 phases, as well as liq...
OTHER CARBON MATERIALS
  fullerenes
                                                      APPLICATIONS
                   ...
Legame σ tra orbitali atomici di tipo np (pz)
Legame π tra orbitali atomici di tipo np (px,py)
L'ibridazione nel carbonio
C (Z = 6)
configurazione elettronica: 1s2 2s2 2p2
                                          1s2...
Ibrido sp3: lobi diretti nello spazio secondo i vertici di un
tetraedro il cui centro corrisponde al nucleo del carbonio

...
Giustificazione dell’orientamento spaziale degli orbitali ibridi sp3


                                                  (...
Ibrido sp2




                    3 elettroni di valenza
                                             1 elettrone di vale...
Ibrido sp




                                           1
                                              (2 s + 2 pz )
   ...
Esempio di molecole con carbonio in stato di
ibridazione sp3: metano
Esempio di molecole con carbonio in stato di
ibridazione sp3: etano
Esempi di ibridi sp2: copresenza di legame σ e π




                                     Etilene,
                       ...
Esempi di ibridi sp2: copresenza di legame σ e π




                           σ
                       Butadiene,

     ...
Esempi di ibridi sp2: copresenza di legame σ e π




                   σ
                             Benzene,

         ...
Esempi di ibridi sp: copresenza di legame σ e π


                                                  Acetilene,
           ...
Struttura del diamante (ibridazione sp3)
ALCANI (ibridazione sp3)

                               butano (n=4)

            metano (n=1)



                 etano ...
Struttura della grafite (ibridazione sp2)




                            Asse c
Vista lungo l'asse c
POLYCONJUGATED MOLECULES

1D π conjugated systems:        2D π conjugated systems:
        polyenes           PAH, Polycyc...
Grafite nanostrutturata
PAH
Polycyclic
Aromatic
Hydrocarbons
Nanotubi di carbonio
(ibridazione prevalente sp2)
Stable forms of carbon clusters: (a) a piece of a graphene sheet,
(b) the fullerene C60, and (c) a model for a carbon nano...
Graphene ribbons terminated by (a) armchair edges and (b)
zigzag edges, indicated by filled circles. The indices denote th...
High-resolution electron
                                     micrographs of graphitic particles


                       ...
Sketch of the cross section of a
PAN carbon fiber along the
fiber axis direction.
Here the in-plane (La) and c-
axis (Lc) ...
Schematic model for the microstructure of activated carbon fibers




                                     Fiber after som...
Nanostructured amorphous carbon films




         D. Donadio, L. Colombo, P. Milani, G. Benedek,
         Phys. Rev. Lett...
Multi-walled carbon nanotubes

                                                      S.Ijima, Nature 358,
                ...
Multi-walled carbon nanotubes
Fullerenes within SWNTs: peapods




                                   La@C82
Heat treatment of peapods produces
double-wall NT
Carbon nanotubes,
M.S. Dresselhaus, G. Dresselhaus,
Ph. Avouris (Eds.) Springer (2001)
(5,5)


(9,0)


(10,5)
Ch = 4 a1+ 2 a2   nanotubo (4,2)
Rosso: 3,3 armchair, θ=45°
Rosso: (5,0) zig-zag θ=0°
The unrolled honeycomb lattice of an Armchair nanotube




                         structural unit


  Ch = Chiral vector...
Electronic 1D density of states per unit cell
of a 2D graphene sheet for two (n,0) zigzag
nanotubes:

(a) the (10,0) nanot...
Derivative of the current-voltage dI/dV
curves obtained by scanning tunnelling
spectroscopy on various isolated single-wal...
nanotube (n,m) → (4,2)
Ch = 4 a1+ 2 a2
Hamiltoniano elettronico H = H(θ1,θ2) alla Hückel
(i.e. tight-binding ristretto a orbitali 2pz)



 (0,-1)                ...
Curve di dispersione elettronica (4,2)



                               π∗
                                              ...
Curve di dispersione elettronica (6,3)
Energia in unità di β




                                                        N...
Curve di dispersione elettronica (17,8)

Energia in unità di β




                            Funzione del numero quantic...
Densità di stati elettronici
                          di due nanotubi chirali
                                  metallici...
Analytic expressions for the electronic energies have been
     obtained with a symmetry treatment of Pz orbitals in the
 ...
(10,10) Ch ≅ 4.2 nm




(10,0) Ch ≅ 2.42 nm
Wave function, Van Hove peak at energy -0.95 Beta units
Tube axis
Energy dispersion and
density of states for
 (9,0) zigzag nanotube



                         Density of states for (150,...
Figure 5: TEM micrographs of seaweed-like carbon
                                                        objects produced ...
Raman spectra of graphite and amorphous carbon
                                                                           ...
A.M. Rao, E. Richter, S. Bandow, B. Chase, P.C. Eklund, K. W.
Williams, M. Menon, K. R. Subbaswamy, A. Thess, R. E. Smalle...
Room temperature RBM spectra for bundles
of SWNTs produced by pulsed
laser vaporization using an Fe/Ni catalyst in a
carbo...
Raman spectroscopy is used to characterize carbon nanotubes;
the G band brings important structural information

         ...
Carbon nanotubes:


extended π-conjugated systems


long range electronic and vibrational interactions


crucial dependenc...
Polyconjugated carbon systems


                                               Polyenes
                      Raman disper...
Modeling electrons and phonons in carbon nanotubes

- Structural unit: 2 atoms     A general treatment for
- Screw axis sy...
Describing the geometry of a generic (n,m) nanotube




          Ch = 4 a1+ 2 a2   (4, 2) nanotube
(14,5)
                         Electronic band structure of
                        semiconducting (4,2) nanotube


     ...
Ohno’s three parameters force field (1) generalised to graphite (2)
              (1)          K. Ohno, J. Chem. Phys. 95,...
Phonon dispersion curves of graphite    S. Piscanec, M. Lazzeri, F. Mauri, A. C.
                                        F...
Generalization of the Ohno Force Field to nanotubes
   of any diameter and chirality

   Method based on graphene cell (2 ...
Bond-bond polarizabilities Πij
        ∂ 2 Eπ      It is directly related to stretching
Π ij ≡
       ∂β i ∂β j    force c...
The G matrix is specific for any given nanotube:

                  G = G(n,m)

                 tube curvature

The F mat...
Raman spectra of individual
                                G band:
    single wall nanotubes
                            ...
Experimental findings                               Empirical force field
     by Jorio et al.                            ...
Dispersion of the G line   Full symbols:   longitudinal phonons
                           Open symbols:   transversal pho...
Phonons of the chiral (6,3) metallic tube
Conclusions


1.   Carbon nanotubes share long range interaction
     physics similarly to other π-conjugated systems
    ...
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
5   P  Nanotubes Chiara
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5 P Nanotubes Chiara

  1. 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. 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. 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. 4. Legame σ tra orbitali atomici di tipo np (pz)
  5. 5. Legame π tra orbitali atomici di tipo np (px,py)
  6. 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. 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. 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. 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. 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. 11. Esempio di molecole con carbonio in stato di ibridazione sp3: metano
  12. 12. Esempio di molecole con carbonio in stato di ibridazione sp3: etano
  13. 13. Esempi di ibridi sp2: copresenza di legame σ e π Etilene, σ H2C=CH2 π
  14. 14. Esempi di ibridi sp2: copresenza di legame σ e π σ Butadiene, H2C=(CH)-(CH)=CH2 π
  15. 15. Esempi di ibridi sp2: copresenza di legame σ e π σ Benzene, C6H6 π
  16. 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. 17. Struttura del diamante (ibridazione sp3)
  18. 18. ALCANI (ibridazione sp3) butano (n=4) metano (n=1) etano (n=2) pentano (n=5) propano (n=3) esano (n=6)
  19. 19. Struttura della grafite (ibridazione sp2) Asse c Vista lungo l'asse c
  20. 20. POLYCONJUGATED MOLECULES 1D π conjugated systems: 2D π conjugated systems: polyenes PAH, Polycyclic Aromatic Hydrocarbons conjugated 2pz orbitals
  21. 21. Grafite nanostrutturata PAH Polycyclic Aromatic Hydrocarbons
  22. 22. Nanotubi di carbonio (ibridazione prevalente sp2)
  23. 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. 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. 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. 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. 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. 28. Nanostructured amorphous carbon films D. Donadio, L. Colombo, P. Milani, G. Benedek, Phys. Rev. Lett., 83, 776-779 (1999)
  29. 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. 30. Multi-walled carbon nanotubes
  31. 31. Fullerenes within SWNTs: peapods La@C82
  32. 32. Heat treatment of peapods produces double-wall NT
  33. 33. Carbon nanotubes, M.S. Dresselhaus, G. Dresselhaus, Ph. Avouris (Eds.) Springer (2001)
  34. 34. (5,5) (9,0) (10,5)
  35. 35. Ch = 4 a1+ 2 a2 nanotubo (4,2)
  36. 36. Rosso: 3,3 armchair, θ=45°
  37. 37. Rosso: (5,0) zig-zag θ=0°
  38. 38. The unrolled honeycomb lattice of an Armchair nanotube structural unit Ch = Chiral vector T = Translation vector (k)
  39. 39. 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).
  40. 40. 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.
  41. 41. nanotube (n,m) → (4,2) Ch = 4 a1+ 2 a2
  42. 42. 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
  43. 43. 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
  44. 44. Curve di dispersione elettronica (6,3) Energia in unità di β NT conduttore Funzione del numero quantico μ = 0 .. 41
  45. 45. Curve di dispersione elettronica (17,8) Energia in unità di β Funzione del numero quantico μ = 0..325
  46. 46. Densità di stati elettronici di due nanotubi chirali metallici Van Hove singularities EF (14,5) (11,8) EF
  47. 47. 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
  48. 48. (10,10) Ch ≅ 4.2 nm (10,0) Ch ≅ 2.42 nm
  49. 49. Wave function, Van Hove peak at energy -0.95 Beta units Tube axis
  50. 50. Energy dispersion and density of states for (9,0) zigzag nanotube Density of states for (150,150) armchair nanotube (150,150) Ch=63 nm
  51. 51. 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.
  52. 52. 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)
  53. 53. 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
  54. 54. 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.
  55. 55. 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)
  56. 56. 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 !
  57. 57. 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
  58. 58. 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
  59. 59. Describing the geometry of a generic (n,m) nanotube Ch = 4 a1+ 2 a2 (4, 2) nanotube
  60. 60. (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
  61. 61. 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
  62. 62. 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
  63. 63. 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
  64. 64. 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
  65. 65. 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)
  66. 66. 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)
  67. 67. 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+
  68. 68. 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
  69. 69. Phonons of the chiral (6,3) metallic tube
  70. 70. 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

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