in the name of allah excitons in single – walled carbon nanotubes nasim moradi graduate student of...
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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