An introduction to x-ray absorption in graphene1

Pourya Ayria

Supervisor: professor Saito

30 March 2013

Overview

Graphene unite cell

The nearest tight binding for graphene

Dispersion relation for 1s orbital

X-ray absorption in graphene

Dipole vector

Graphene unit cell

graphene

(From chowdhury thesis)

Brilloin zone

1 2

3 1 3 1, , , ; 3 2.46

2 2 2 2o

c ca a a a a a A

The nearest tight-binding for graphene

The electronic dispersion of graphene

The nearest tight-bindig for graphene

The electronic dispersion of graphene without considering overlap S=0

23( ) 1 4cos cos 4cos

2 2 2y yxk a k ak a

k

*

1 1

( ) ( )( ) ; ( )

2 ( )(1 ( )) 2 ( )(1 ( ))C CA B

f k f kC k C k

k s k k s k

23( ) 3.033 1 4cos cos 4cos

2 2 2y yxk a k ak a

E k

Eigenvalue

Eigenvalue

/ 3 /2 3( ) 2 cos( )2

x x yik a ik a k af k e e

The calulated energy dispersion relation of 1s band of graphen

1s orbital energy -283.5 is considered under fermi level; however, some papares indicated that it is around -285. and t=0.1, S=0.

X-ray absorption in graphene:

Dipole approximation:

Matrix element for optical transition:

are the tight binding wave function Ip the energy density of electromagnetic wave. D dipole vector. P polarization of wave.

The transition probability per one second as function of k:

The absoption itensity I(E) : is the density of states at energy E

2, ,

1[ ] ( ); ( ).

2 opt Absorption Emission

ieH i eA V r H A t

m m

0

( , ) ( ) ( )

( , ) exp( ( ) ) ( , ).

fi fopt final initial f opt i

fiopt f i f i f i

M k k k H k

IeM k k i t D k k P

m c

,i f

22 4

2

2 3 20

sin ( ( ) ( ) )4 2

( , ) . ( , )( ( ) ( ) )

f if i x

fif i f i f i

x f i x

E k E k Ee I

W k k P D k km c E E k E k E

2 42

2 30

4( ) . ( , ) ( )fi

f ix

e II E P D k k E dk

m c E

( )E

Dipole vector

Dipole vector for transition to (matrix.f90)

The oscillation strength is

1s *

* *

* * * 2

( , ) ( , ) ( , )

ˆ( , ) ( ) ( ) ( ) ( ) 0.3

ˆ( , ) ( ) ( ) ( ) ( ) ( ) ( ) 5.2 10

on offf i f i f i

on f i f if i A f A i B f B i

off f i f if i A f A i B f B i

D k k D k k D k k

D k k C k C k C k C k z

D k k C k C k f k C k C k f k z

*( , ) ( , ) ( , )if iff i f i f iO k k D k k D k k

Reference:

Physical Properties of Carbon Nanotues, R.Saito,G.Dresselhaus,M.S Dresselhaus, Impreial college Press

M.T.Chowdhury M.S Thesis

Thanks dear hesky and nugraha.

END