Materials Sciences and Applicatio ns, 2010, 1, 72-76
doi:10.4236/msa.2010.12013 Published Online June 2010 (http://www.SciRP.org/journal/msa)
Copyright © 2010 SciRes. MSA
Ferroelectric Phase Transition in Graphene with
Anderson Interaction
Natalia Yanyushkina1, Mikhail Belonenko2, Nikolay Lebedev1
1Volgograd State University, Volgograd, Russia; 2Volgograd Institute of Business, Laboratory of Nanotechnologies, Volgograd, Russia.
Email: yana_nn@inbox.ru
Received February 7th, 2010; revised April 20th, 2010; accepted April 22nd, 2010.
ABSTRACT
The normal transverse electric field which appears in impurity graphene spontaneously in the presence of a high applied
electric field was calculated. The given effect can be associated with non-equilibrium of electron subsystem in graphene.
The characteristics of spontaneous field on the parameters of the problem were investigated.
Keywords: Non-Equilibrium Phase Transition, Araphene, Anderson Model
1. Introduction
A study of phase transitions is the one of the famous
paradigm in modern fundamental physics. It should be
noted that phase transition (in wide extend) means a sub-
stance’s transition from one phase to another at the change
of external conditions-temperature, pressure, magnetic
field and electric field, and so on. And phase transition, in
restricted sense, is saltatory variation of physical proper-
ties at the continuous changing of the external parameters.
So, in particularly, non-equilibrium phase transition takes
place one of the main in variety of phase transitions. This
kind of transition arises at the presence of external fields
with the different characters.
In [1,2] it was shown theoretically that under the action
of a high electric field non-equilibrium phase transitions
are possible in the electron gas in conductors with a
body-centered cubic lattice. The effect consists in spon-
taneous appearance of a transverse field playing the
role of order parameter. The applied electric field ,
directed along one of the crystal symmetry axis, is the
controlling parameter. A necessary condition for the ap-
pearance of a transverse field is the non-additivity of the
electron energy spectrum when the electron energy:
y
E
x
E
),()()()( zyx pppp


where p
is the quasi-
momentum of electrons (e.g., when using a tight-binding
approximation the electron spectrum in the body-centered
cubic lattice becomes non-additive:
, where а is the lattice constant.
Besides, the spectrum must be bounded.
x
os(p a(p)c
/ 2)
yz
cos(pa / 2)cos(pa / 2)
All these conditions are carried out for the impurity gra-
phene, and one can investigate an existence possibility of
phase transitions in impurity graphene, which is considered
in the frameworks of the Anderson model. It can be expect
the appearance of transverse component x when a filed
E
y
E is applied (which plays the role of order parameter).
Graphene is a structure which consists of one layer of car-
bon atoms, located in the units of hexagonal lattice. A great
attention is paid to large electron mobility in the graphene
and to its unique properties which are an alternative of silicic
base in the modern microelectronics [3-6]. We note that
electromagnetic waves in the carbon structures become
strongly nonlinear even in the weak fields that gives rise to
spread possibility of solitary electromagnetic waves in the
graphene and carbon nanotubes. These properties of carbon
nanostructures have theoretical interest and attempts of
applying in the nonlinear optics [7]. Nonlinearity is caused
by change of classical function of electron distribution and
by non-parabolic dispersion law of electrons.
Summarizing, one can draw a conclusion, that the prob-
lem of graphene response in the magnetic field with taking
into account Anderson interaction between the impurity
and graphene electrons, is very important and actual.
2. Basic Equations
Let us consider the response of graphene on external electric
field along axis x in geometry is given in Figure 1.
Then the Anderson Hamiltonian of the electron system
can be written in the Formula (1) [8,9].
where ,,,
j jjj
aabb

are the creation and annihila- tion
operators of electrons in a graphene unit with spin , t
is
the overlap integral between adjacent grapheme units
determined by overlapping of the wave functions of the
Ferroelectric Phase Transition in Graphene with Anderson Interaction 73
Figure 1. Geometry of a problem
grapheme electrons, U is the constant of Coulomb repul-
sion of the electros at the same unit; are the
creation and annihilation operators of impurity electrons
in a graphene spin;
+
jσjσ
d,d
is the energy of impurity electrons;
U1 is the energy of Coulomb repulsion of impurity elec-
trons only. V is the hybridization parameter.
The following parameters were estimated by MNDO
method [10]: 2 eV, 12 eV, 12 eV, V
t
U1
U
2
eV. Since the properties of the model described by the
Anderson Hamiltonian is sufficiently complex, further
we will use the approximation: U→∞ and consider, all
mean value is to be spatially homogenous. Note that the
approximation U→∞ is well supported by quantum
chemical calculations for graphene-like. In this approxima-
tion the Hamiltonian (1) can be written as [11,12].
()kk
k
H
Eps k CC

(2)
where

kk
C,C
are the creation and annihilation op-
erators of elementary excitations (with momentum k and
spin σ), and is the spectrum which, according to
[11,12], can be presented as:
)k(Eps


2
2)1(4))((
)(
2
1
)(
Vnnk
nkkEps
im


(3)
where V is the hybridization parameter, )( k
is the
spectrum of graphene electrons determined by the Ham-
iltonian , а and are parameters determined
from the self-consistency conditions of the problem.
h
H
nim
n
Note, that dispersion law, which describes graphene
properties without taking into account Coulomb repulsion
at the same unit, has a following form [13].
y
2
x
ap ap
(k)14cos(ap)cos( )4cos( )
33
 y
(4)
where 7.2
eV, a3b/2
,
b
0.142 nm is the
distance between adjacent carbon atoms in graphene, and
),yx p(pk
. According to [11,12], the self-consistency
conditions are:
We use the average electron method with taking into
1
()(
()
()
himhyb
hjjjjjjjjjjjj
jj
imjj jjjjjj
j
hybjjj j
j
HH HH
Ht abbaUaaaabbbb
HddddUdddd
HVadda
 
  
 
   

 
 
 
 



)
(1)
vv
k,v
im
vv
k,v
2
v
v
vv
2
v2im
v
2
2im
nAF(E(k)),
nBF(E(k)),
V
A(1) ,
2Q (k)
E(k)
B(1) ,
2Q (k)
1
E(k)n(1)((k)n)4(1n)V
2
1
Q(k)((k)n)4(1 n )V
2




 





 



,
(5)
Copyright © 2010 SciRes. MSA
74 Ferroelectric Phase Transition in Graphene with Anderson Interaction
account the motion equation can be written as [14].
dp =E
dt (6)
one can apply:1A
Ect

, and for low-temperaturecase it
is possible to obtain:
x
0x x
y0yy
p
=p +Et
p
=p +Et
It should be noted:
() ()
(), ()
x
y
xy
x
Epsp, pEpsp, p
υpυp
p


xy
y
p
Further, we use the average electron method, according
to which the current can be expressed as [15].
(())exp()
¥
0
j= υpt -tdt
(7)
where j is the density of current, )(
t
pis the solution of (6)
with some initial conditions, which correspond to the
energy minimum. In our case, it is necessary to consider
the solutions of (8) for four initial conditions (correspond
to minimum of ):
)( pE
03
xxy y
ap=;ap= π;ap= π/;ap=π23
/
.
Then we should sum up all values for current.
It is convenient to represent the dispersion law of gra-
phene in the following form:
2
() cos()cos()
1()cos()cos()
(2 )
xymn xy
m,n
ππ
mnx yxyxy
-π-π
εp,p=Ampnp ,
A= εp,pmpnpdpdp
π

Finally, in this case:
222 22
22
cos(2)cos() (-1)
33
(-1- )
(1())(1() )
i
xmn
m,n
xy x
yx yx
mk
j= Aπ+π××
mE nEmE
×+nE+mE+nE-mE

(8)
The transverse field is defined by boundary condi-
tions for the given applied filed . Let us assume that a
circuit is opened in the x-direction:
x
E
y
E
0
x
j (9)
This condition corresponds to some solve for the trans-
verse field: . Equation (9) has two solves:
)( yxx EEE
0
))(1)()(1(
)1(
0
,
22
22222


nm xyxy
xy
mn
x
mEnEmEnE
EmEnm
A
E
(10)
The transverse field spontaneously appears in one of
two mutually opposite directions at the some values of
parameters in the second equation of (10). In this case, we
deal with non-equilibrium one-order phase transition. The
appearance of the transverse field component represents
perhaps the simplest example of self-organization in the
impurity graphene.
3. Calculation Results
A typical dependence of on , which described by
(8), is represented in Figure 2. Dependence of the trans-
verse field on the applied field , which deter-
mined as non-zero solve of (10) is given in Figure 3 and
Figure 4.
x
jx
E
x
Ey
E
It worth to note that analogous dependence on the pa-
rameter V is weak, and existence of two non-zero solves is
more important. One of this solve (smaller in module) is
thermodynamically unstable. As might be expected, ac-
cording to the numerator of expression (10) increase in the
field leads to an increase in the field , due to the
y
Ex
E
non-additivity of the electron dispersion. Note the fact that
the dependence of the field on is mainly asso-
x
Ey
E
ciated with the view of the electron spectrum without
interaction (4), and the effects of the interaction of elec-
trons with impurities contribute only relatively small
corrections. This can also be attributed to the effect is due
to non-additivity of the dispersion and less sensitive to the
particular type of spectrum. It should be noted that it fol-
lows from (10) and Figures 3 and 4 at large can be
y
E
written: yx EE
that may be useful in evaluating the field
intensities in the experiment. Dependence of a minimal
value of the field , at which the transverse field appears,
on the at the different is given in Figure 5.
y
E
im
n
n
Note, that the minimal value of is more strongly
depends on impurity concentration. It can be associated
with reconstruction of the graphene electron spectrum in
the presence of impurities.
y
E
The transverse field
x
E, which is emergent sponta-
neously, can be thermodinamically unstanble, as opposite
to always stable solve for open circuit in the x-direction
= 0.
x
E
We also investigate the stability using the method pro-
posed in [1]. We introduct the following function:
fixedE
tconsEdEjEФ
y
E
xxxx
x

0
tan)()( (11)
The given function is usually called with synergetic
potential and it is the analogue of a thermodynamic
potential for non-equilibrium tasks. According to [1], the
Copyright © 2010 SciRes. MSA
Ferroelectric Phase Transition in Graphene with Anderson Interaction 75
Figure 2. Dependence of current density on the field Ex,
when the field Ey is fixed (Ey = 4.0). All magnitudes are in the
non-dimensionless units
Figure 3. Dependence of the field Ex on the field E
y:(a)
; (b) . All magni-
tudes are in the non-dimensionless units
25.0,1.0nim 
n25.0,5.0nim 
n
stable conditions are:
00 2
2

x
xdE
Фd
dE
dФ (12)
These formulas mean that in given non-equilibrium
situation the function (11) attains its minimum in the
stationary state. Thus the function Ф may be regarded as
the analogue of a thermodynamic potential for equilibr-
ium systems. Dependence of the “potential” Ф on the field
for some values of the is presented in Figure 6.
It can be seen that the function Ф has the minimum and the
maximum. It should be noted, maximum corresponds to a
smaller in module solve of Equation (10), and minimum
corresponds to a larger in module solve. It means the
x
Ey
E
Figure 4. Dependence of the field on the field :(a)
; (b) . All magni-
tudes are in the non-dimensionless units
x
E
,1
n
y
E
25.0,1.0n im 
n75.0.0nim 
Figure 5. Dependence of the minimum on the:
(a) ; (b); (c) . All magnitudes
are in the non-dimensionless units
y
Eim
n
25.0
n50.0
n75.0
n
larger in module solve is stable. Note that dotted branhes
in Figure 3 and Figure 4. correspond to the maximum of
function Ф (unstable solve), but solid curve and dashed
curve correspond to minimum (stable solve).
This transition, in which the electric field appears
spontaneously, is concerned to ferroelectric type. Though,
the transverse field plays the role of the order pa-
rameter, and the field
x
E
y
E
is the analogue of temperature
(controlling parameter).
The transverse field occurred spontaneously, obviously,
can be found by measuring the charge on the capacitor,
Copyright © 2010 SciRes. MSA
76 Ferroelectric Phase Transition in Graphene with Anderson Interaction
Copyright © 2010 SciRes. MSA
Figure 6. Dependence of the function Ф on the field Ex, when
the field Ey is fixed: (a) Ey = 3.5; (b) Ey = 4.5; (c) Ey = 5.5. All
magnitudes are in the non-dimensionless units
which is must be attached to the ends of graphene sheets,
oriented in the x direction. A transverse field will con-
tribute to accumulation of charge on the capacitor plates,
and the total charge will be clearly defined potential dif-
ference, which creates an electric field necessary to com-
pensate for a transverse electric field.
4. Conclusions
In conclusion we formulate our main results:
1) The appearance of the electric field, which is per-
pendicular to the external applied field, in impurity The 1.
The appearance of the electric field, which is perpen-
dicular to the external applied field, in impurity graphene
with the Anderson interaction was obtained.
2) The minimal value of the applied field is strong
defined by electron concentration in the impurity.
3) The analysis of the synergetic potential has shsown
that the emergent state with the spontaneous transverse
field is stable.
5. Acknowledgements
This work was supported by the Russian Foundation for
Basic Research under project
No. 08-02-00663 and by the Federal Target Program
“Scientific and pedagogical manpower” for 2010-2013
(project NK-16(3)).
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