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
Brilloin zone
 3 1
 3 1
a1  a 
,  , a2  a 
,   ; a  3ac c  2.46A o
2
 2 2
 2
(From chowdhury thesis)
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

Eigenvalue
 3k x a 
k a
 k ya 
2 y 
E (k )  3.033 1  4 cos 
cos

4
cos





 2 
 2 
 2 
 3k x a 
k a
 k ya 
2 y 

4
cos
 cos 



 2 
 2 
 2 
 (k )  1  4 cos 
f (k )  e ik x a /

3
 2e ik x a /2 3 cos(
k ya
2
)
Eigenvalue
f (k )
f * (k )
C
C (k ) 
;C B (k )  
2 (k )(1  s1 (k ))
2 ( k )(1  s1 ( k ))
C
A
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:
H 
1
ie
[i   eA ]2 V (r ); H opt ,Absorption ,Emission 
A (t ).
2m
m
Matrix element for optical transition:
fi
M opt
(k final , k initial )   f (k f ) H opt  (k i )
fi
M opt
(k f , k i ) 
e
m 
I
c 0
exp(i (f  i   )t )D (k f , k i ).P
,f
iare
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:
 

sin 2 (E f (k f )  E i (k i )  E x ) 
2 

W (k f , k i ) 
P .D fi (k f , k i )
2 3
f
i
2
 0 m c E x
(E (k f )  E (k i )  E x )
4e 2 4 I 
2
The absoption itensity I(E) :
 (E ) is the density of states at energy E
I (E ) 
4e 2 4 I 
 0 m 2c 3E x

2
P .D fi (k f , k i )  (E )dk
Dipole vector

Dipole vector for transition 1s to  *
D (k f , k i )  D on (k f , k i )  D off (k f , k i )

, k )  C

(matrix.f90)
D on (k f , k i )  C Af (k f )C Ai (k i )  C Bf (k f )C Bi (k i )  0.3zˆ
D off (k f

i
*
f *
A
*

(k f )C Ai (k i )  f (k )  C Bf (k f )C Bi (k i )f * (k )  5.2 10 2 zˆ
The oscillation strength is
O (k f , k i )  D *if (k f , k i )D if (k f , k i )
*
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