Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice
Low-temperature thermodynamic properties of two-dimensional electron ensemble on ordered and disordered host-lattice is studied numerically by transfer-matrixes method. It is shown that at low temperatures and weak disordering of host-lattice sites positions change in chemical potential leads to suc...
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irk-123456789-1169882017-05-19T03:02:46Z Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice Slavin, V.V. Низкоразмерные и неупорядоченные системы Low-temperature thermodynamic properties of two-dimensional electron ensemble on ordered and disordered host-lattice is studied numerically by transfer-matrixes method. It is shown that at low temperatures and weak disordering of host-lattice sites positions change in chemical potential leads to successive transitions of the system from ordered phases (like generalized Wigner crystal) to disordered states. The ranges of stability of these crystals as the function of temperature, chemical potential and disorder parameter are established. 2009 Article Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice / V.V. Slavin // Физика низких температур. — 2009. — Т. 35, № 2. — С. 197-203. — Бібліогр.: 18 назв. — англ. PACS: 05.10.–a, 05.20.–y http://dspace.nbuv.gov.ua/handle/123456789/116988 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Slavin, V.V. Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice Физика низких температур |
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Low-temperature thermodynamic properties of two-dimensional electron ensemble on ordered and disordered host-lattice is studied numerically by transfer-matrixes method. It is shown that at low temperatures and weak disordering of host-lattice sites positions change in chemical potential leads to successive transitions of the system from ordered phases (like generalized Wigner crystal) to disordered states. The ranges of stability of these crystals as the function of temperature, chemical potential and disorder parameter are established. |
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Slavin, V.V. |
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Slavin, V.V. |
| title |
Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice |
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Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice |
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Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice |
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Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice |
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Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice |
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low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы |
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http://dspace.nbuv.gov.ua/handle/123456789/116988 |
| citation_txt |
Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice / V.V. Slavin // Физика низких температур. — 2009. — Т. 35, № 2. — С. 197-203. — Бібліогр.: 18 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT slavinvv lowtemperaturethermodynamicsoftwodimensionalelectrongasondisorderedhostlattice |
| first_indexed |
2025-07-08T11:25:10Z |
| last_indexed |
2025-07-08T11:25:10Z |
| _version_ |
1837077797034524672 |
| fulltext |
Fizika Nizkikh Temperatur, 2009, v. 35, No. 2, p. 197–203
Low-temperature thermodynamics of two-dimensional
electron gas on disordered host-lattice
V.V. Slavin
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: slavin@ilt.kharkov.ua
Received November 13, 2008
Low-temperature thermodynamic properties of two-dimensional electron ensemble on ordered and disor-
dered host-lattice is studied numerically by transfer-matrixes method. It is shown that at low temperatures and
weak disordering of host-lattice sites positions change in chemical potential leads to successive transitions of
the system from ordered phases (like generalized Wigner crystal) to disordered states. The ranges of stability
of these crystals as the function of temperature, chemical potential and disorder parameter are established.
PACS: 05.10.–a Computational methods in statistical physics and nonlinear dynamics;
05.20.–y Classical statistical mechanics.
Keywords: Generalized Wigner crystal, thermodynamics, low-dimensional systems, disordered systems.
1. Introduction
The layered and low-dimensional conductors attracts
significant interest of researchers. This interest caused by
a number of unusual thermodynamic and kinetic proper-
ties inherited by such systems. Among these systems the
conductors with Coulomb self-localization (CS) are of
special interest. It is well known that CS appears when
charge carriers tunneling between host-lattice sites is sup-
pressed by their mutual Coulomb repulsing. The charge
carriers becomes fully localized on host-lattice sites and
the only mechanism of particles motion is jumps of elec-
trons (or holes). As the result, the dynamics of CS sys-
tems is discrete. The exact criterion of CS is the smallness
of overlapping integral, t, in contrast with the typical
change �� �� ( )a /l0
2 in inter-particle repulsion energy as
it hops (tunnels) between the host-lattice sites. Here a0 is
mean host-lattice spacing, l a /ce
d� 0
11( ) / mean inter-par-
ticle distance, � the average Coulomb energy per particle,
d dimension of the system and ce is electron concentra-
tion. As was shown in [1], in zero-temperature limit and
in the case of ce �� 1, the charge carriers forms ordered
structure — generalized Wigner crystal (GWC). The
ground state (GS) structure of GWC is fully described in
terms of one-dimensional (1D) theory developed by Hub-
bard [2–4]. It should be noted that GWC formation has
universal nature irrespective of the potential of inter-par-
ticle repulsion details and geometry of host-lattice. Such
a «lowering of dimension» leads, in particular, to rather
specific zero-temperature dependence of ceagainst chem-
ical potential, �, which is well developed fractal structure
of «devil staircase» type [3].
One of the most famous examples of 2D GWL is the so
called MOSFET structure (metal-oxide-semiconductor
field-effect transistor) with holes impurity band. Besides,
the systems where the charge carriers are pressed out on
surface by external electric field applied perpendicular to
surface also belong to CS class of conductors. In this case
the applied electric field plays the role of chemical poten-
tial �. Changing in � varies electron concentratio ce in
wide range. Besides, there are strong reasons to suggest
that CS criterion (t � ��) can be also fulfilled in layered
metalooxides of high temperature superconductor types.
Another important group is 1D CS compounds. These
are, first of all, quasi-one-dimensional organic conductors
[4]. Furthermore, a lot of artificially created 1D nano-sys-
tems systems, such as chains of quantum dots, exchanging
by electrons [5], chains of metallic nano-grains with tunnel
junctions (organic molecules of different types) between
them [6] also belong to this group.
Is of special interest the question about an influence of
disorder in host-lattice positions on low-temperature
properties of these systems. This question appears natu-
rally because the overwhelming majority of CS conduc-
tors are essentially disordered systems. For instance, in
© V.V. Slavin, 2009
semiconductors MOSFET type this disorder is caused
by random distribution of impurities [7,8], in some
nanostructures [5,6] the disorder is determined by a dis-
persion on tunneling junctions.
Besides analytical approaches to such system investi-
gations (see, for example [1–4,9,10]), an interest to the
numerical studies has grown rapidly during the last de-
cade. This interest is caused by the fact that the analytical
theories are mainly focused on 1D systems and based on
very rough simplifying models that leads, as the result, to
qualitative estimations only. At the same time just the de-
tails of the ground state structure, the exact structure of
elementary excitations spectrum, the thermodynamic and
kinetic properties of such systems contain the most inter-
esting and important information. This information, in
particular, casts light on the extremely significant ques-
tions about the ergodic or non-ergodic behavior of the
systems, the spectrum of relaxation times etc. In this con-
nection the methods of computer simulations like
Monte-Carlo [11,12] become the most powerful and use-
ful scientific instruments. The merits and demerits of the
methods are well known. The universality and flexibility
should be attributed to advantages of these methods. At
the same time the difficulties accompanying the investi-
gation of non self-averaging characteristics of the sys-
tems (for example, the configurations) are shortcomings,
undoubtedly. Besides, the accuracy of calculation drops
crucially with disorder growing, whereas the calculation
time increase essentially. Last time new modifications of
Monte-Carlo methods have appeared. First of all, this is
the so-called multicanonical approach [13]. However, the
questions about convergence and authenticity of the ob-
tained results are open yet. As opposed to Monte-Carlo
simulation, we propose exact computer investigation of
2D CS systems using transfer-matrix method [14]. The
evident merit of this method is easy controlled accuracy.
The demerit is the restriction on the system size caused by
computer productivity. The matter is that the calculation
time for this model is proportionate to L L2 3 , where L is
linear size of the system. Analogously, the memory size,
required for the calculation � 22L.
The main goal of this paper is to study low-tempera-
ture thermodynamic properties of 2D CS systems and in-
fluence of disorder in host-lattice sites positions on these
properties. Besides, even in the case of regular host-lat-
tice the theory developed in [1] is inapplicable for con-
centration range1 2 1/ ce� � . The proposed method allows
us to fill this gap.
2. Hamiltonian
We will consider rarified electron gas (ce � 1) and,
thus, spin indexes are dropped (model of spin-less fer-
mions). Besides, we will neglect tunneling of electrons
between host sites. The Hamiltonian of such a system, H ,
can be written as:
H � �
1 2/ u n n n
i j
i j i
i
i j(| | )
r r
r r r
r
r – r � , (1)
here ri are coordinates of 2D host-lattice sites; independ-
ent variable ni � 0 1, is a number of electrons in ri th
host-lattice site (occupation numbers); �is chemical po-
tential and u i j(| | )r r� a screened Coulomb potential of the
inter-electron repulsion; summation is carried out over all
the sites. It is clear that in the case of ordered host-lattice
the vectors ri can be written as r a bi k l� � , where a and b
are primitive translation vectors and k, l are some integer,
in the case of diordered host-lattice the vectors ri ar-
ranged randomly.
3. Approximation
In this paper the lattice-gas model with near-neighbor
(NN) approximation is used. In spite of the simplicity,
this model allows us to establish new properties of the
system under consideration. Adaptation of the model de-
mand an additional punctuality, because the long-range
potential of inter-electron repulsion is cut off over the dis-
tances a0 (we will consider squad host-lattice with inter-
val between host-lattice sites equals a0). At the same
time, it is well known that the details of the u r( ) do not af-
fect qualitatively on the thermodynamic characteristics
[2,3,15], if the following restrictions should be fulfilled:
(i) u r( ) is monotonic, everywhere convex function, (ii)
u r( ) diminishes faster than r �1. NN-approximation does
not satisfied the above restrictions in full extend, but as
was recently shown in the case of 1D systems of these sort
[10], such changes in u r( ) lead to a weak modification of
the characteristics of the system only. Besides, we will
consider strong-screened Yukava-like potential with ra-
dius of interaction R a0 0� :
u r u r /R r(| | ) exp( | | )| |� �0 0 ,
constant u 0 was chosen as the following: u a /R0 0 0� exp( ),
so that u a( )0 1� for all R0. All the mentioned above al-
lows us to consider the proposed model as completely ad-
equate one.
As usual, instead of Hamiltonian (1) with random
host-lattice sites positions and regular function u r(| | ) it is
convenient to transfer randomness into potential of inter-
action and consider the system as an electron ensemble on
regular host-lattice with random function u r(| | ). In such
an approximation the Hamiltonian (1) has the following
form:
H � �
� �
u n n ni j i
i j
N
j i
i
N
,
,
� ,
(2)
198 Fizika Nizkikh Temperatur, 2009, v. 35, No. 2
V.V. Slavin
here N is the total number of host-lattice sites; all these
sites are enumerated by index i u i j, , is random energy
of interaction between the sites with numbers i and j, sign
<i, j> means that the summation in first term is taken over
near neighbors of ith site. In the simplest case of strictly
ordered squad host-lattice the Hamiltonian (2) can be eas-
ily represented in spin variables s ni i� 2 1– , where
si � �11, .
H � �
�
�
�
� �
~ ( ~ ) ( ~ )
,
u
s s
u
s
u N
i j
i j
N
i
i
N
4
2
2 2
� �
. (3)
Here ~ ( ) ( )u u a u a� �0 0 2 . It should be stressed that in
Hamiltonian (3) the inverse-symmetry is broken for all
� 2 0
~u . In other words, beside «external filed» �, an ef-
fective «internal» field determined by ~u affects on
«spins». That is why in the limit �
0 the ground state of
(2) and (3) is vacuum-like one.
In this paper we will consider squad host lattice and,
thus, a i b j� �a a0 0, (i and jare unit vectors in X and Y di-
rections, correspondingly). It is convenient to rewrite the
microscopic variables n
ir as the following: n n
i k lr � , ,
where r a bi k l� � (k,l are integer).
The interaction energies u i i(| | )r r� � we rewrite as the
following:
u i i(| | )'r r� �
� � � � � � � � � � �� �u k k l l u k l k lk l k l(| ( ) ( ) )| ) ( , , , ), ,a b � � .
Here � k l, and � � �k l, are random shift vectors of the
correponding host-lattice sites. The module of � has nor-
mal distribution law with dispersion � which is varied
from 0 (ordered host-lattice) up to 1 (complete disorder).
The angular dependence of � is random one with uniform
distribution. The Hamiltonian (2) acquires the form:
H � � �
�� � �
�
� �
�
�
l
L
k
L
l l
L
k k
L
k lu k l k l n n
11 1
1
1
1
( , , , )
''
, k l, � �
�
��
� nk l
l
L
k
L
,
11
.
(4)
Here L is linear size of the system (L N2 � ). We impose to-
roidal boundary condition on the system. It means that
n n n nk L i L l l, , , ,,� �� �1 1 1 1 .
Let us introduce a set of occupation number values
�-th line
�� � � �� { , , , }, , ,n n nL1 2 � .
Taking into account the toroidal boundary condition
� �L � �1 1. In NN-approximation �th line interacts with
( )� �1 th and ( )� � 1 th lines only. The interaction energy of
�th and ( )� � 1 th lines, E( , )� �� � � 1 , has the form
E u k k n n
k
k k( , ) ( , , , ) , ,� � � �� � � �� � �
1 1 +
� � � �
� �u k k n n
k
k k( , , , ) , ,� � � �1 1 1 1
� � �
� �u k k n n
k
k k( , , , ) , ,� � � �1 1 1 1. (5)
The first term in (5) is interaction energy of electrons in �th
line with vertical neighbors from ( )� � 1 th line, second and
third terms are corresponding energies of diagonal interac-
tions. Besides, let us introduce E( )�� which is energy of
electron interaction in �th line (horizontal interaction) plus
the energy produced by chemical potential �.
E u k k n n n
k
k k k l
k l
( ) ( , , , ) , , ,
,
� � � �� � �� � �
�1 1 .
(6)
The corresponding transfer-matrixes �P� [14] have the form:
� � ��
� ��
� �� � �
� � �� � �
| � |
( ( , ) ( ))
P T
E E
1
1
1
e .
Here T is the temperature. It should be noted again that the
proposed method allows us to include into consideration
both vertical (horizontal) and diagonal NN interactions.
The big sum is
Z
H
� �
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�e Spur
=1
N
T
ni
P
{ }
�
�
�
, (7)
the summation is carried out over microscopic states { }ni .
Thermodynamic potential
f T
N
T( , ) ln ( )� � �
1
Z (8)
has been calculated numerically. For algorithm testing the
dependence f(T,0) has been calculated for microscopic
variables ni � –1,1 (Ising model) and ordered host-lat-
tices 6�6, 8�8 and 10�10. The result is presented in
Fig. 1. The exact solution (dashed curve) is plotted for
comparison. One can see good agreement even for rather
small host-lattices.
4. Low temperature thermodynamics
As was shown in [1] the ground state configuration of
2D lattice electron gas on ordered host-lattice is fully de-
scribed in term of 1D Hubbard’s theory [4]. It means, in
particular, that for any fixed � the ground state configura-
tion is periodic structure (crystal) composed of one or two
sort of electron bands. Both these bands are parallel, infi-
nite in one direction and have fixed shift vector in other
direction. If the concentration of the bands is of the form
c /q qe ( ) , , , ...� � �1 2 3 , then the only one sort of electron
bands appear in the GS structure (see, for example,
Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice
Fizika Nizkikh Temperatur, 2009, v. 35, No. 2 199
Fig. 2,a). Otherwise, two types of bands form the ground
state structure (see Fig. 2,b). The mutual arrangement of
these bands can be calculated on the base of simple for-
mula [2]:
r a i/ci e� 0[ ]. (9)
This formula was obtained for 1D systems of this sort. It
describes the mutual arrangement of the electrons with
concentration ce over 1D host-lattice with period a0. Sign
[...] means an integer part, ri is the position (coordinate)
of ith electron. In 2D case ri has sense of the position of
the electron «band» with number i. As mentioned above,
one of the most interesting consequence of such structure
of the GS is zero-temperature dependence ce ( )� which is
well-developed fractal structure of devil-stair type [2,3].
In this connection it is reasonable to start the investiga-
tion of 2D systems from low-temperature dependence
ce ( )� . It’s also interesting and important, because the
results of theory [1] are applicable for concentration re-
gion c /e � 1 2 only. At the same time the proposed NN ap-
proximation allows us to investigate the thermodynamic
properties of the system under just in the region ce � 1.
As far as c
N
ne i
i
�
�
1
1
, the low-temperature depen-
dence ce ( )� can be calculated as the following:
c
f T
e T( )
( , )
|�
�
�
� �
�
�
0.
The results of calculation ce ( )� for � � 0 (ordered
host-lattice), a0 1� and fixed temperature T u a� �10 3
0( )
are presented in Fig. 3. One can see stair-like dependence
with the stairs, corresponding to rational electron concen-
trations. In the limiting case of R a0 0�� , when the only
vertical and horizontal NN interactions are taken into
considerations, the stairs with ce = 1/2 and 1 (curve a) ap-
pear. Taking into account diagonal NN interactions leads
to appearance of additional stairs with ce = 1/4 and 3/4
(curve b). It means that the effective lowering of the di-
mension and generalized Wigner crystal (GWC) forma-
tion, discovered in [1] preserves in the region1 2 1/ ce� � .
An influence of the disorder in host-lattice positions, �
on ce ( )� dependence is presented in Fig. 4. The calcu-
lat ions were carr ied out for host- lat t ice 10�10,
T u a� �10 3
0( ) and R0 1� . Besides, an additional averag-
ing of the results over 10 random realizations were car-
ried out. The dependence of typical fluctuation � as the
function of realization number N 0 is presented in Fig. 5.
As it seen from the figure, an increase of N 0 over 10 does
not effect on � almost.
200 Fizika Nizkikh Temperatur, 2009, v. 35, No. 2
V.V. Slavin
0 2 4 6 8 10
–7
–6
–5
–4
–3
–2
0 0.5 1.0 1.5 2.0 2.5
–2.2
–2.1
–2.0f
T
10 10�8 8�
Exact solution
6 6�
Fig. 1. The results of calculation f T( , )� � 0 for regular Ising
model (ni � �11. ). Host-lattice sizes are: 6�6, 8�8 and 10�10.
In addition, the curve corresponding to exact solution is pre-
sented. The region of maximal deviation numerical and exact
solutions is presented on insert.
a b
Fig. 2. Examples of 2D GWL corresponding to different elec-
tron concentrations ce. c /e �1 3 (a), c /e � 2 5 (b).
0.2
0.4
0.6
0.8
1.0
c e
a b c
0 1 2 3 4 5 6 7
Fig. 3. The dependence ce( )� for ordered host-lattice (� � 0)
and different values of R T0 � � �10 3
0u a( ), host-lattice size is
10�10. R0: 10
–3
(a), 0.25 (b), 1 (c).
5. Phase transitions
In this paragraph we will consider the phase diagram
of the system. As far as Hamiltonian (3) has «anti-
ferromagnetic» nature due to inter-electron repulsion, the
most appropriative method is studying c Te ( ) dependence
at different �. This dependence is analogue of magnetiza-
tion M over T dependence at different values of external
magnetic fields H for magnetic systems. As it well
known, the extremum point corresponds to phase transi-
tion temperature (N�el point). Typical dependences c Te ( )
are presented in Fig. 6. One should pay attention to the
structures of these curves. As was mentioned above, in
the simplest case of ordered host-lattice the system can be
considered as spin assemble placed in both «internal
field», produced by «exchange» interaction u and «exter-
nal field» conditioned by �. These fields have different
sign and at some values of � � 2~u the total «field»
changes the sign. Besides, there is additional term
– ( )�N/2 in Hamiltonian. That is why at � � 2~u the de-
pendence c Te ( ) changes their geometry. For � � 2~u ,
c T / M Te ( ) ( )� �1 2 and c T / M Te ( ) ( )� �1 2 for � � 2~u .
Here M T( ) as a function which is analogous of M T( ) for
antiferromagnets. For � � 2~u the total «field» equals zero
and c T /e ( ) � �const 1 2.
Determining extremal points of c Te ( ) dependence for
different � one can plot the dependence Tc ( )� , where Tc is
phase transition temperature. The results of numerical
calculation for ordered (� � 0) and disordered (� � 0 05. )
host-lattices are presented in Fig. 7. One can see that for
each region of � in zero-temperature devil-staircase there
is an «hat» in Tc ( )� dependence. It means that for any fi-
nite temperature T change in � parameter transforms the
system from ordered phases (GWC) to disordered ones
(like Wigner glass). The critical temperature is maximal
in the vicinities of � interval centers, tending to zero on
the borders. This phenomenon explains a number of ex-
periments in 2D semiconductor layers in external perpen-
dicular electric fields. This field plays the role of �
because press out the volume electrons to 2D layer. For
example, in experiments of M. Pepper group [16,17]
change in external field leaded to drastic change in elec-
tric conductivity in the layer. It means that in some inter-
val of external electric fields (or in some �-regions in
terms of proposed model) the GWC forms. The conduc-
tivity of the system decreased crucially due to localiza-
tion of electrons.
Low-temperature thermodynamics of two-dimensional electron gas on disordered host-lattice
Fizika Nizkikh Temperatur, 2009, v. 35, No. 2 201
0 1 2 3 4 5 6 7
0.2
0.4
0.6
0.8
1.0
m
c e
Fig. 4. An influence of disorder parameter � on low-tempera-
ture dependence ce( )� for T u a� �10 3
0( ), R0 1� and for host-lat-
tice size 10�10.
2 4 6 8 10 12
0.002
0.004
0.006
0.008
0.010
0.012
0.014
� = 0.05
N0
0.016
Fig. 5. Typical fluctuation � as the function of initial configu-
rations number N0 for two values of disorder parameter �
(� � 0.01 and � � 0.05).
0 1 2 3 4
0.35
0.40
0.45
0.50
0.55
T
c e
1
2
3
4
Fig. 6. Series of the dependences c Te( ) corresponding to dif-
ferent � values on ordered host-lattice (� � 0). Host lattice size
is 10�10, R0 1� . �� 3.5 (1), 3 (2), 2.5 (3), 2 (4).
As it seen from Fig. 6, the heights of these «hats» de-
crease with disorder parameter � growth. This allows us
to establish the critical temperature Tc as the function of
disorder parameter �. T Tc c
max ( . )� �� 3 1 and R0 1� is pre-
sented in Fig. 8. The chemical potential � � 3 1. corre-
sponds to the maximum of Tc for devil-staircase step with
c /e �1 2. Solid line in this figure corresponds to the best
fitting by the function T Ac � �0 0( )� � � . Critical disor-
der parameter � 0 � 0.123, power parameter � � 0.38.
6. Results and discussion
We have shown that the model of lattice gas with
near-neighbor interaction leads to a rich thermodynamics.
In spite of the simplicity of the model, it contains the most
essential features of discrete 2D systems of such a sort. It
is established that in low-temperature limit and weak
host-lattice sites position disorder the system under con-
sideration is generalized Wigner crystal. It means, in par-
ticular, that zero-temperature dependence of electron
concentration ce as the function of chemical potential � is
fractal structure of devil staircase type. In this structure
for each interval of � there is a rational value of ce
(so-called devil stair). This result fully confirms to pre-
liminary investigations [1], which are valid, however, for
concentration range c /e � 1 2. Now, one can to assert that
the generalized Wigner crystal formation takes place in
whole concentration range ce � 1.
It is clear, that at any finite temperature T and disorder
parameter � the devil stairs are «diffused». The ranges of
stability and corresponding critical temperatures Tc as the
function of chemical potential and disorder parameter
have been established. Besides, it is shown that change in
chemical potential leads to successive transitions of the
systems from ordered phases (generalized Wigner crys-
tal) to disordered states like Wigner glass. This phenome-
non may explain experimental results where small change
in electron concentration leads to giant oscillation of con-
ductivity in 2D plains (see, for example, [16–18]). In-
deed, in this experiments the concentration of electrons
(or holes) was varied by external electric filed applied
perpendicular to 2D plain. As was mentioned above, this
field plays the role of chemical potential. As the result,
the conductivity in 2D plain changes drastically due to
transitions from ordered phases to disordered states. It
should be noted, that the question about conductivity
properties of such systems is far from this paper frame-
works. Indeed, we neglect the processes related to finite-
ness of t. Such an approximation is absolutely reasonable
in «dielectric» (GWL) phase due to inequality t �� �� (see
Introduction), but the finiteness of t becomes important in
the conductive («metallic») phase, where self-screening
effects becomes strong.
It is planned to study the kinetic properties of such sys-
tems taking into account finiteness of t by Monte-Carlo
methods in near future.
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0
1 2 3 4 5 6 7 8
0.1
0.2
0.3
0.4
T
c
T
max
c
c = 1/4e c = 1/2e c = 3/4e
c = 1e
Fig. 7. The dependence Tc( )� obtained for ordered (� � 0) and
disordered (� � 0.05) host-lattice size 10�10 and R0 1� .
0.00 0.04 0.08 0.12 0.16
0
0.1
0.2
0.3
0.4
T
c
Fig. 8. The dependence T Tc c
max (� �� 3.1) as the function on
�. Solid boxes are the results of numerical calculation. Solid
line id the approximation by the function T Ac � �0 0( )� � �.
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