2 20

0

22

0

22

Recall the Drude theory:

( ) 1( ) , where

( ) 1 1

1 4(in simple metals, )

4 4

4where plasma frequency.

In Graphene the density of states at the Fermi level

p

p p

J t ne ne

E t m i i m

ne

m

ne

m

0

vanishes, so n should be small.

However, is not well known but surely very large.

So we can see that the problem of evaluating is not trivial.

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.

M. Lewkowicz and B. Rosenstein, PRL 102, 106802 (2009)Dynamics of Particle-Hole Pair Creation in Graphene find:

2 2

1disorder 0 0

4lim lim 1.27

e e

h h

2 2

20 disorder 0

lim lim 1.57 2

e e

h h

They support this value of the dc conductivity of pure graphene

2

3

e

h Other authors find

There is no accepted theoretical value

1

1

=1.7

Previous experiments on Si substrates give 4

There is no accepted experimental value

They report

Measurement of

conductivity

4

0( )

We study the time-dependent transport through Graphene using

1ˆ( ) [ ( )] ,1

ˆ ( ) in Heisenberg picture (M.Cini 1980)

mn mn H

mn

J t Tr fJ t fe

J t

2.46 Angstromsa

semiinfinitesemiinfinite

0( )

After sudden switching of voltage we calculate the current

1ˆ ˆ( ) [ ( )] , ( ) in Heisenberg picture.1

( )Conductivity (t)=

.

mn mn mnH

mn

J t Tr fJ t f J te

J t

V

2 0|d E

J

E

1 0|d V

I

V

2

0

e

h

*

vF

Lat t t

6

2

2

first plateau =2

4second plateau = independent of geometry

e W

h L

e

h

8

Fullerenes

Sir Harold W. Kroto, University od Sussex,Nobel Prize for Chemistry in 1996

Discovery September 4,1985

Fullerenes consist of 20 hexagonal and 12 pentagonal rings as the basis of an icosohedral symmetry closed cage structure.

Was known initially as soccerene

20 hexagons 12 pentagons 32

20 hexagons 12 pentagons 20*6 12*5 180 2 90

0 Euler : 2 60

F

S S

g F S V V

9

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.

In theory, an infinite number of fullerenes can exist, their structure based on pentagonal and hexagonal rings, constructed according to rules for making icosahedra.

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.

11

Endohedral compounds

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

12

The art of hitting the goal with every shot

http://www.univie.ac.at/qfp/research/matterwave/c60/index.html

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.

13

Carbon Nanotubes

Fascinating electronic and mechanical

Properties:1. 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

14

15S. Iijima. "Helical microtubules of graphitic carbon." Nature 354 56 (1991)

Multi-Walled NanoTube(MWNT)

16

17

Carbon Nanotubes

S. Iijima, Nature 354, 56 (1991)

18From Wikipedia

Carbon Nanotubes: Lattice Structure

1919

Carbon Nanotubes: Lattice Structure

Graphene sheet Nanotube

S. Iijima, Nature 354, 56 (1991)

L≈1md≈nm

Single Wall ( , ) CNT

with , primitive lattice vectors,

roll-up-vector of ( , ) CNT

( atom at is identified to theone at ( ) ( ) )

Depending on n,m theCNT can be metal or semiconductor

mn

a b

na mb n m

xa yb x n a y mb

20

This is a possible choice of the basis which is often used:

1a

2a

Then, (n,0) nanotubes are called zigzag nanotubes, and (n,n) nanotubes are called armchair nanotubes. Otherwise, they are called chiral.

1 2

00

( , ) translation in Graphene

identified with ( , ) nanotube.mn

mn

m n R ma na

R R m n

1 2| | | | 3b a a a

21

1a

2a

(4,0) zigzag CNT

(15,0) zigzag CNT

axis

of

CN

T

path towards the tip

path around the belt: 2n atoms

(n,0) alias Zigzag CNT

22

1a

2a

(2,2) armchair CNTarmchair CNT

path along the y axis

All armchair nanotubes are metallic, as suggested by paths along axis

armchair CNT

CNT axis

= y axis

2323

• “Chiral” geometry•all the rest

“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

24

1 2 1 2

3 3 3 3( , ), ( , ), 32 2 2 2

a a a a a a a

The alternative basis which we used for the band structure of Graphene is also in use for CNT

1a

2a

1 2

00

3 3In this basis, ( , ) translation ( ( ), ( ))

2 2( , ) : identified with .

nm

nm

n m R na ma a n m n m

n m CNT R R

Since both conventions are used we must be ready to handle both of them.

path around the belt: 2n atoms

Zigzag CNT

25

1a

2a

Zigzag CNT using alternative basis

(4,0) zigzag CNT becomes

(4, 4) in new basis

The is na 3circumference

1 2 1 2

3 3 3 3( , ), ( , ), 32 2 2 2

a a a a a a a

,

Indeed, the shift along the belt of CNT is obtained with ,

(0, 3) .n n

m n

R a n

CNT axis

= x axis2

3

3a

aa

The wavevector around belt is quantized: 3 2 , integer

3The CNT radius

2

k k na

na

26

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

2

0 0

0 0

3 3 3( ) 1 4cos cos 4cos .

2 2 2

2, 0, 1, 2,

3

3cos cos and one finds, setting (along axis)

2

3band structure: ( , ) 1 4cos c

2

x y y

y

y x

E k E J k a k a k a

k kna

k a k qn

E q E J qa

2

os 4cosn n

Since the wavevector around belt is quantized:

3 2 , integer, just insert into Graphene

band structure

yk k

k na

2727

Carbon Nanotubes as quasi 1D systems: one component of k quantized

• NT: Compact transverse dimension Discretization of k

• Band Structure of graphene

Subbands correspondto differentvalues of k

k|| is a continuous

variable k||

28

2

0 0

Note:

3band structure: ( , ) 1 4cos cos 4cos

2

integer quantum number, (n,n)=chirality index

dependence along x: c

E q E J qan n

3os since the distance along x between two A atoms

2

3 3is . Setting c=

2 2

qa

a a

31 with ,

2d BZ q c a

c c

Note also: ( , ) ( , )E q E q

29

for 6, 0, 1, 2,3 each with 12 bands ,

0,3 nondegenerate since cos( 3)=0.6

n E

Note: the (4,-4) zigzag CNT has 8 atoms around the

belt. Generally,

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

2

0 0( , ) 1 4cos cos 4cos

3, . Since cos cos ( , ) ( , )

2

E q E J qcn n

BZ q c a E q E qc c n n

Examples: (n,-n) zigzag: for 5, 10 bands: 0, 1, 2

each occurs twice because of double sign in ;

3 2no more bands since 3 yields cos( ) cos( ).

5 5

n

E

30

2

0 0

0 0

0 0

3The dispersion ( , ) 1 4cos cos 4cos ,

2

3implies two nondegenerate bands ( ,0) 5 4cos ,

2

and for even , , two flat bands ( , )2 2

E q E J qc c an n

E q E J qa

n nn E q E J

2

1 cos( ) . Therefore :

3 21

cos 3 and 1, 6 and 2,.... if 3* | | ; then,2

radicand

Zigzag CNT are Metals fo

1 4cos cos 4cos

r n multipleof 3.

2(1

if n if n nn

qcn n

2

0 0

0

cos ).

1 cos( )Using cos , 2 | cos( ) |

2 2 2

at the border of the BZ, , for both bands. No gap!

qc

qcE E J

qc E E

31

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

32

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

33

1a

2a

(2,2) armchair CNT armchair CNTpath along the axis

All armchair nanotubes are metallic, as suggested by paths along axis

armchair CNT

CNT axis

= y axis

34

2a

Recall Primitive vectors

1 2

Cartesian components are :

3 3 3 3( , ), ( , ),2 2 2 2

a a a a

1

1cos( )

3 2

1 2

1 2 1 2 1 2 1 2

angle too; indeed,3

3 3 3 30 *3 | | | || | | || | sin( )

2 2 2 2 3

3 30

2 2

a a

i j k

a a k a a a a a a

35

(2,2) armchair CNT

Length of belt:6a

Armchairs are (n,n) using basis 1a

2a

1 2

00

1 2

Nanotubes: the site at

3 3( ( ), ( ))2 2

is identified

orthogonal vector axis

(( ), 3( ))

mn

mn

R ma na a n m n m

R

CNT

T ma na n m n m

,

taking m=n, .

(3 ,0),

(belt of CNT along x axis),

is quantiz

y

e

axis = CNT axis

d 3 2

along

n n x

y

armchair

R a n k k

k k an

q k

CNT axis

= y axis

36

2

0 0

3 3 3Graphene : ( ) 1 4cos cos 4cos

2 2 2

2for armchair CNT: set , , 0, 1, 2,

3

3 3 3cos cos cos( ),

2 2 2

3 3 2cos cos

2 2 3

x y y

y x

y

x

E k E J k a k a k a

k q k kna

k a qa qc c a

k a ana

20 0

cos

( , ) 1 4cos cos 4cos

n

E q E J qc qcn

0 0 0

0,

2 2( ,0) [1 2cos ] cross at qc , ( ,0)

3 3armchair CNT are always metals.

For

E q E J qc E E

37

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