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**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.

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**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

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Measurement of conductivity

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semiinfinite

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**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

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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

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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

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

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**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

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**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

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**Multi-Walled NanoTube**

(MWNT) S. Iijima. "Helical microtubules of graphitic carbon." Nature (1991)

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

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**Carbon Nanotubes: Lattice Structure**

From Wikipedia

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**Carbon Nanotubes: Lattice Structure**

S. Iijima, Nature 354, 56 (1991) L≈1m d≈nm Graphene sheet Nanotube 19

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**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.

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

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**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

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**“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

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**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

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**Zigzag CNT using alternative basis**

CNT axis = x axis 25

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

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**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

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**Note: the (4,-4) zigzag CNT has 8 atoms around the belt. Generally,**

(n,-n) zigzag 2n atoms in belt 2n bands

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**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

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**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.

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**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

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**Recall Primitive vectors**

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**Armchairs are (n,n) using basis**

CNT axis = y axis

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

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