IN THE NAME OF ALLAH

EXCITONS IN SINGLE –

WALLED

CARBON NANOTUBES

NASIM MORADIGRADUATE STUDENT OF

ATOMIC AND MOLECULAR PHYSICS

UNDER SUPERVISION OF :

DR. FAZELI AND DR. MOZAFFARI

QOM UNIVERSITY

outline

i. Introduction to Carbon’s Structures

ii. Structure of Carbon Nanotubes

iii. Excitons

iv. Bethe - Salpeter Equation

Introduction to

Carbon’s Structures

Until the mid-1980’s pure solid carbon was

thought to

exist in only two physical forms : diamond and

graphite.

CARBON The Carbon atom has six electrons . .

1 s  𝟐   2s  𝟐   2 p  𝟐

Graphite

Diamond

In 1985 , Richard Smalley and group of researchers

made an interesting discovery :

Nanotubes

Graphene

Structure Of Carbon Nanotubes

Carbon nanotubes were discovered in 1991 by Iijima.

Graphite A single layer of graphite, graphene

a carbon nanotube made of a single graphite layer rolled up into a hollow cylinder

with diameter as small as nm

Length: few nm to microns

Animation from S. Maruyama’s carbon nanotube site

multi-walled nanotube (mwnt)Diameter ~ 10 – 50 nm

Single -walled nanotube (swnt)

Diameter ~ 0.5 - 2nm

J.Charlier and X. Blase ,” electronic properties of nanotubes” , Rev . Mod . Phys .79 ( 2007 ).

IMAGES OF NANOTUBES

Chiral vector

lrsm.upenn.edu

SWNT’s geometry

specified by a pair of

integers (n , m)

a = lattice constant of the honeycomb networka = ( , the C-C bond length)

diameter

𝑑𝑡=¿ 𝑐h∨¿/𝜋=𝑎𝜋

√𝑛2+𝑛𝑚+𝑛2 ¿

𝐶h=𝑛𝑎1+𝑚𝑎2

Tube axis

Rev . Mod . Phys.79 ( 2007) , p:680

Chiral angle () = angle between and .

(n , 0) (= 0) zigzag(n , n) (= 30) armchair

(n , mn0) chiral

www.nanodic.com

BONDING

bonds with threenearest carbon atoms

Graphiticc-c bonds c=c 152 Kcal/mole

Tight – binding model The tight-binding model , we imagine how the wave functions

of atoms or ions will interact as we bring them together.

𝝍 𝑨 𝝍𝑩

𝝍 𝑨+𝝍𝑩 𝝍 𝑨−𝝍𝑩

1( ) exp( . ) ( )k m m

m

r ik rN

Bloch function :

¿1

exp( .( ))n m m nm n

k H k ik HN

exp( . )k nn

E k H k ik

Charles Kittel , introduction to solid state physics , ( Wiley ,1983)¿

Tight – binding model of graphene

Rev . Mod . Phys .79 , p:684

𝝋𝑨(𝒌 ,𝒓 )=𝟏

√𝑵∑𝒍

¿¿

(𝑯𝑨𝑨 𝑯 𝑨𝑩

𝑯𝑩𝑨 𝑯𝑩𝑩)

𝑯 𝑨𝑩=𝟏𝑵∑

𝒍 , 𝒍 ′𝒆¿ ¿¿

𝑯 𝑨𝑨=𝑯𝑩𝑩=𝟎

�⃗�𝟏=𝒂( √𝟑𝟐

,𝟏𝟐 )

�⃗�𝟐=𝒂( √𝟑𝟐

,−𝟏𝟐 )

20

3( , ) 1 4cos cos 4cos

2 2 2y yx

x y

k a k ak aE k k

¿𝜶 (𝒌)=𝟏+𝒆− 𝒊𝒌 .𝒂𝟏+𝒆− 𝒊𝒌 .𝒂𝟐 ( −𝑬 𝑯 𝑨𝑩

𝑯𝑩𝑨 −𝑬 )𝑬 ±(𝒌)=±𝜸𝟎√¿¿

= 2.9 eV

ethan minot , Tuning the band structure of cnt , PhD Thesis , cornell univ (2004 ) , paper : 28

Periodic boundary conditions along the circumferential direction

𝝍𝒌(𝒓 +𝒄𝒉)=𝒆𝒊𝒌 .𝒄𝒉𝝍𝒌(𝒓 )  =𝝍𝒌(𝒓 )

From the Bloch theorem

. 1hik ce 𝑘⊥=2𝜋 𝑙

¿𝑐h∨¿¿

(7,7) (7,0)

ethan minot , PhD Thesis , cornell univ (2004 ) , paper : 32

K

1 2( ) / 3K b b ⃗⃗⃗

. 2hK C l⃗⃗

1 2hC na ma ⃗

. 2i j ija b

3n m l

3n m l p

ethan minot , Tuning the band structure of cnt , PhD Thesis , cornell univ (2004 ) , paper : 33

Metallic nanotubes

3n m l ( : an integer)l

Semiconducting nanotubes

3 1n m l

For a (5,5) Armchair

Electronic band structure

Density of states

Rev.Mod.Phys . 79 , p: 686

For a (10,0 ) zigzag

An energy gap opens at

DOS have a zero value at the fermi energy .

Rev.Mod.Phys . 79 , p: 687

Exhibits a metallic behavior.

In semiconducting zigzag or chiral nanotubes the Band gap is independent of the chiral angle and :

For a (8,2 ) Chiral

102 /g cc tE a d

Rev.Mod.Phys . 79 , p: 688

applications

Electrical1. Capacitors

2. Diodes and transistors

3. Flat panel displays

4. Data storge

Energy storage1. Lithium batteries

2. Hydrogen storage

Biological1. Bio-sensors

2. Functional AFM tips

3. DNA sequencing

Optical properties1. Solar cells

2. Quantum information processing

3. Optical communication

Quantum cryptographyCarbon nanotubes could be used as a source of single photons for

applications in quantum cryptography.

Ch.Galland ,et al , “ Photon Antibunching in the Photoluminescence Spectra of a

Single Carbon Nanotube”, Phys. Rev. Lett. 100, 217401 (2008).

Credit: Grossman/Kolpak

Optical Properties And

Excitons

An exciton is a bound state of an electron and hole which are attracted

to each other by the electrostatic coulomb force.

Grosso , Solid state physics , paper : 233 ex g bE E E

exciton ” exciton ” was introduced by Frenkel in 1931 .

Frenkel Wannier - Mott Charge transfer

Andre Moliton , solid state physics for electronics , paper : 364

exciton

No charge

S = 0 , 1 [singlet , triplet]

Boson

Bright and Dark

Bethe – Salpeter

Equation

Bethe – Salpeter Equation

Hans Bethe1906 - 2005

Nobel Prize for Physics (1967)

Edwin Ernest Salpeter

1924 - 2008

[ ] n eh n n nck vk vck v c k vck

v c k

E E A vck K v c k A A

The equation was actually first

published in 1951.

describes the bound states of a two-body

(particles) system in a formalism.

C.Spataru , S.Ismail Beigi “ Excitonic Effects of SWNT ” , Phys.Rev.Lett.92 , ( 2004 )

Original article : A Relativistic Equation for Bound-State Problems

E.E.Salpeter and H.Bethe , Phys.Rev. 84 , 1232-1242 (1951)

[ ] n eh n n nck vk vck vck vck

v c k

E E A v c k K vck A A

0BSE ehH H K BSE n n nH

n nvck

vck

A vck

* *, , , , ( ) ( ) w(r ,r) ( ) ( )direct

k c k c k v k vK drdr r r r r *

, , , , ( ) ( ) v(r ,r) ( ) ( )xk c k v k c k vK dr dr r r r r