THE QUANTUM KINETIC EQUATIONS FOR THE SYSTEM,
«EPITAXIAL GRAPHENE - QUANTUM FILM»
Z.Z. Alisultanov1
1
Prokhorov General Physics Institute Russian Academy of Sciences, Moscow, Russia
ANNOTATION
We consider the problem of epitaxial graphene, which formed on the quantum film. We
give a microscopic derivation of quantum kinetic equations for a system of « graphene –
quantum film».
Introduction
Due to its unique properties, graphene is attracting the attention of many researchers in
various fields of science and technology. Recently, more attention is paid to the study of the
electronic states of graphene formed on the surface of metals and semiconductors [1-3]. Of
particular interest is the study of the electronic states of graphene formed on the surface of
low-dimensional systems. In this paper, based on the method of quantum Green's functions
in the Kadanoff-Baym formalism, a microscopic derivation of quantum kinetic equations
for a system of «graphene – quantized film» .
The quantum kinetic equations
The Hamiltonian of the system is Hˆ  Tˆ  Uˆ  Vˆ , where Tˆ the single-particle
Hamiltonian of the system, Û - the interaction operator with a variable external field, Vˆ
- the potential of interparticle interaction. In accordance with [4], we introduce the nonequilibrium Green's function
 
  

g
(1;
2),
g
(1;
2) 

g (1; 2) 
.
  
  
 g (1; 2), g (1; 2) 
Here

1  ( z1r1 , t1 )


- a set of spatial and temporal coordinates.
 
Notation
 
1  (r1 , t1 ); 1  ( z1 , r1 , t1 ) . g (1; 2), g (1; 2) - Green's function of electrons of graphene and
the substrate, respectively, and the rest of the Green function describing the exchange
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processes between the electron density of graphene and the substrate. The equation of
 
motion for electron Green's function of grapheme
 
g (1; 2), g (1; 2) has the form


 
 
ˆ (1)  u (1) g (1, 2) 
i

h
g


 t1


(3)
 


  
  
  d 3 (1, 3)   P (1, 3) g (3, 2)   (1 2),




 
where  P (1, 2) - the contribution due to the interaction with the substrate
 
L
 L
0
0

 
 
 
 P (1, 2)    d 3   d 4  (1,3) g 0 (3, 4) (4, 2),
(4)
 
where g 0 (1; 2) the electron Green's function of the substrate in the absence of interaction
 
with the graphene. In the future, it is convenient to notations g (1; 2)  G 1;2 ,
 
 
 
 (1, 2)  (1;2) for graphene, g (1; 2)  g (1;2) ,  (1, 2)   (1,2) for substrate. After
the transition to real time arguments have [5]


 ˆ 
  




h
(
1
)

u
(
z
,
r
,


)
G
(
z
r
;
z
g
1
1
1
1
2 , r2 ;  , ) 

2
2 
d R
  
  dr3 
(   RP )( r1 ; r3 ; 
,  ) 
2
2


d
 G (r3 ; r2 ;  ,   )   dr3 

2
2



  

 (  P )( r1 ; r3 ; 
, ) G A ( r3 ; r2 ;  ,   ),
2
2
(7)
here
L
 R , A
L



 P (r , r ;  )   d z1  dr1  d z 2  d r2 (r; z1 , r1 ;  ) 
0
 R , A


0

 g 0 ( z1 r1 ; z 2 , r2 ; ) ( z 2 r2 ; , r ; ),
To determine the effective wave equation for the electrons of graphene and the substrate,
we represent the mass operator
transformation:
 P [ g0 ]
in the form of two terms with the
g 0  g  ( g 0  g ) :  P g 0    P g    P g 0  g  .
~
~
The term  P
we take into account when determining the effective wave equation of the electron
graphene. Another part of the  P reserve in the equation for the correlation function.
Thus, the


~
hˆg (r ) q (r ,  )   dr1 Re  R   P (r; r1 ;  ) 
  q (r1 ,  )  E (q,  ) q (r ,  ),
(8)
where q - a two-dimensional electron wave vector of graphene, which describes the
motion of electrons in graphene. Finally, we obtain the quantum kinetic equation for the
electron subsystem of grapheme
k

k
 



E
(
q

,


)

E
(
q

,


) G ( q, k ;  , ) 

2
2
2
2 
k

k
 

 u (q  ,   ) u (q  ,    G (q, k ;  , ) 
2
2
2
2

dk d R
k  k
  
R
(



)(
q

,


; k , ) 
P
 (2 ) 3
2
2

k

dk d
 G (q  ,   ; q  k ,   )  

2
2
2

k  k
  
 G (q 
, 
; k , ) 
2
2
k

 ( A   PA )( q  ,   ; q  k ,   ) 
2
2


dk d
k  k
  

(  P )( q 
, 
; k , ) 
3
2
2
(2 )
k

dk d
 G A (q  ,   ; q  k ,   )  

2
2
(2 ) 3
k  k
  
 G R (q 
, 
; k , ) 
2
2


k

 (  P )( q  ,   ; q  k ,   ).
2
2
The equations of motion for the functions G nRn( A ) take the form

k

k
  R ( A)

  2  E (q  2 ,   2 )  u (q  2 ,   2 )  Gnn ' (q, ; k , ) 
d  R ( A)
k  k
  

( nn   PR ,(nnA) )(q 
, 
; k , ) 
(11)
2
2
2
k

GnR(nA' ) (q   ,  ; k  k ,   )  1,
2
2
Equation (10) together with a similar equation for the electron size-quantized film allows us
to study both equilibrium and nonequilibrium properties of the «graphite – quantized film».
For non-equilibrium phenomena can be attributed, such as fluctuations in the electron
density of epitaxial graphene under the influence of an external electromagnetic field
(plasma oscillations), zero sound, etc.
References
[1] A.H. Castro Neto, F. Guinea, N.M.R. Peres et al., Rev. Mod. Phys. 81,109 (2008)
[2] S.Yu. Davydov, Technical Physics Letters 37, 64 (2011)
[3] Z.Z. Alisultanov, R.P. Meilanov, Physics of the Solid State, 2012 (in press)
[4] R.P. Meilanov, Journal of Surface Investigation. X-ray, Synchrotron and Neutron
Techniques 3, 52 (1999)
[5] D.C. Langreth, P. Nordlander, Phys. Rev. B 43, 2541 (1991).