1. Graphene conductivity
A lot of effort has been devoted to the question of transport in pure graphene due to the remarkable fact that the dc conductivity is finite without any dissipation process present.

2. M. Lewkowicz and B. Rosenstein, PRL 102, 106802 (2009)
Dynamics of Particle-Hole Pair Creation in Graphene find:
They support this value of the dc conductivity of pure graphene
Other authors find

3. Measurement of
conductivity

4. 4 4

5. semiinfinite

6. 6 6

8. Fullerenes Discovery September 4,1985 Was known initially as soccerene
Sir Harold W. Kroto, University od Sussex,Nobel Prize for Chemistry in 1996
Discovery September 4,1985
Was known initially as soccerene
Fullerenes consist of 20 hexagonal and 12 pentagonal rings as the basis of an icosohedral symmetry closed cage structure. 8

9. In theory, an infinite number of fullerenes can exist, their structure based on pentagonal and hexagonal rings, constructed according to rules for making icosahedra.
Il fullerene non è molto reattivo data la stabilità dei legami simili a quelli della grafite ed è inoltre ragionevolmenteinsolubile nella maggioranza dei solventi. I ricercatori hanno potuto aumentare la reattività fissando dei gruppi attivi alla superficie del fullerene. 9

10. per produrre i fullereni: arco elettrico, a circa 5300°K, con una corrente elevata e bassa tensione, utilizzando elettrodi in grafite in atmosfera inerte (argon) a bassa pressione. 10

11. Endohedral compounds
They are fullerene cages with La or other metal atoms inside. Some have been crystallized and found to superconduct 11

12. The art of hitting the goal with every shot
We have observed de Broglie wave interference of the buckminsterfullerene C60 with a wavelength of about 3 pm through diffraction at a SiNx absorption grating with 100 nm period. This molecule is the by far most complex object revealing wave behaviour so   far. The buckyball is the most stable fullerene with a mass of 720 atomic units, composed of 60 tightly bound carbon atoms. 12

13. Carbon Nanotubes Fascinating electronic and mechanical Properties:
Depending on their chiralities, nanotubes can be metallic, semimetallic or semiconducting
2. Remarkably high Young’s moduli
and tensile strength
“Imagine the possibilities: materials with ten times the strength of steel and only a small fraction of the weight!”
------Former resident Bill Clinton

15. Multi-Walled NanoTube
(MWNT)
S. Iijima. "Helical microtubules of graphitic carbon." Nature (1991)

17. Carbon Nanotubes
S. Iijima, Nature 354, 56 (1991)

18. Carbon Nanotubes: Lattice Structure
From Wikipedia

19. Carbon Nanotubes: Lattice Structure
S. Iijima, Nature 354, 56 (1991)
L≈1m
d≈nm
Graphene sheet
Nanotube 19

20. This is a possible choice of the basis which is often used:
Then, (n,0) nanotubes are called zigzag nanotubes, and (n,n) nanotubes are called armchair nanotubes. Otherwise, they are called chiral.

21. (n,0) alias Zigzag CNT
axis of CNT
path towards the tip
path around the belt: 2n atoms

22. armchair CNT path along the y axis
CNT axis
= y axis
path along the y axis
All armchair nanotubes are metallic, as suggested by paths along axis

23. “Chiral” geometry “Armchair” geometry (n,m) with m=n, always metallic
“Zig-zag” geometry
(n,m) with m=0 e.g.
(5,0),(6,4),(9,1) are semiconducting
“Chiral” geometry
all the rest 23

24. The alternative basis which we used for the band structure of Graphene is also in use for CNT
Since both conventions are used we must be ready to handle both of them.
Zigzag CNT
path around the belt: 2n atoms

25. Zigzag CNT using alternative basis
CNT axis
= x axis 25

26. pz Electronic bands of (n,-n) zigzag CNT-tight-binding approximation

27. Carbon Nanotubes as quasi 1D systems: one component of k quantized
Band Structure of graphene
NT: Compact transverse dimension Discretization of k
Subbands
correspond
to different
values of k
k|| is a
continuous
variable 27
k|| 27

29. Note: the (4,-4) zigzag CNT has 8 atoms around the belt. Generally,
(n,-n) zigzag  2n atoms in belt  2n bands

31. From Mahan’s nutshell book : band structure of a (5,0) zigzag nanotube
From Mahan’s nutshell book : band structure of a (5,0) zigzag nanotube. Labels indicate angular momentum a values

32. From Mahan’s nutshell book : band structure of a (6,0) zigzag nanotube
From Mahan’s nutshell book : band structure of a (6,0) zigzag nanotube. Labels indicate angular momentum a values. If m-n is a multiple of 3 the nanotube is metallic.

33. armchair CNT path along the axis
CNT axis
= y axis
path along the axis
All armchair nanotubes are metallic, as suggested by paths along axis

34. Recall Primitive vectors

35. Armchairs are (n,n) using basis
CNT axis
= y axis

37. From Mahan’s nutshell book : band structure of a (5,5) armchair nanotube. Labels indicate angular momentum a values. All armchair nanotubes are metallic.