Exciton condensation in quantum wells
The theory of exciton condensation is given in two-dimensional systems under suggestion that condensation occurs in really space and condensed phase arises as a result of an attractive interaction between excitons. Due to the finite value of exciton lifetime the sizes of exciton condensed phase r...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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Sugakov, V.I. 2017-06-13T09:23:36Z 2017-06-13T09:23:36Z 2006 Exciton condensation in quantum wells / V.I. Sugakov // Физика низких температур. — 2006. — Т. 32, № 11. — С. 1449–1457. — Бібліогр.: 30 назв. — англ. 0132-6414 PACS: 73.21.Fg, 78.67.De https://nasplib.isofts.kiev.ua/handle/123456789/120895 The theory of exciton condensation is given in two-dimensional systems under suggestion that condensation occurs in really space and condensed phase arises as a result of an attractive interaction between excitons. Due to the finite value of exciton lifetime the sizes of exciton condensed phase regions are restricted and the condensed phase appears in a form of system of islands amid exciton gas. The joint solution of kinetic equations for island size and exciton diffusion equation in the space between islands has been obtained. The theory is applied to explanation of experimental manifestation of condensed phase in quantum wells and also to explanation of the periodical fragmentation, which was observed in luminescence spectrum from a ring around a laser spot in a crystal with double quantum wells. For such explanations the theory does not require the exciton Bose–Einstein condensation. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Quantum Effects in Semiconductors and Insulating Materials Exciton condensation in quantum wells Article published earlier |
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Exciton condensation in quantum wells |
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Exciton condensation in quantum wells Sugakov, V.I. Quantum Effects in Semiconductors and Insulating Materials |
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Exciton condensation in quantum wells |
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Exciton condensation in quantum wells |
| title_fullStr |
Exciton condensation in quantum wells |
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Exciton condensation in quantum wells |
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exciton condensation in quantum wells |
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Sugakov, V.I. |
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Sugakov, V.I. |
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Quantum Effects in Semiconductors and Insulating Materials |
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Quantum Effects in Semiconductors and Insulating Materials |
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2006 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The theory of exciton condensation is given in two-dimensional systems under suggestion that
condensation occurs in really space and condensed phase arises as a result of an attractive interaction
between excitons. Due to the finite value of exciton lifetime the sizes of exciton condensed
phase regions are restricted and the condensed phase appears in a form of system of islands amid
exciton gas. The joint solution of kinetic equations for island size and exciton diffusion equation in
the space between islands has been obtained. The theory is applied to explanation of experimental
manifestation of condensed phase in quantum wells and also to explanation of the periodical fragmentation,
which was observed in luminescence spectrum from a ring around a laser spot in a crystal
with double quantum wells. For such explanations the theory does not require the exciton
Bose–Einstein condensation.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120895 |
| citation_txt |
Exciton condensation in quantum wells / V.I. Sugakov // Физика низких температур. — 2006. — Т. 32, № 11. — С. 1449–1457. — Бібліогр.: 30 назв. — англ. |
| work_keys_str_mv |
AT sugakovvi excitoncondensationinquantumwells |
| first_indexed |
2025-11-26T15:19:29Z |
| last_indexed |
2025-11-26T15:19:29Z |
| _version_ |
1850626180998234112 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11, p. 1449–1457
Exciton condensation in quantum wells
V.I. Sugakov
Institute for Nuclear Research, 47 Nauki Ave., Kiev, 03680, Ukraine
E-mail: sugakov@i.com.ua
Received May 24, 2006, revised July 7, 2006
The theory of exciton condensation is given in two-dimensional systems under suggestion that
condensation occurs in really space and condensed phase arises as a result of an attractive interac-
tion between excitons. Due to the finite value of exciton lifetime the sizes of exciton condensed
phase regions are restricted and the condensed phase appears in a form of system of islands amid
exciton gas. The joint solution of kinetic equations for island size and exciton diffusion equation in
the space between islands has been obtained. The theory is applied to explanation of experimental
manifestation of condensed phase in quantum wells and also to explanation of the periodical frag-
mentation, which was observed in luminescence spectrum from a ring around a laser spot in a crys-
tal with double quantum wells. For such explanations the theory does not require the exciton
Bose–Einstein condensation.
PACS: 73.21.Fg, 78.67.De
Keywords: two-dimensional systems, exciton condensation theory, luminescence spectrum periodical
fragmentation.
1. Introduction
There were many attempts to observe exciton con-
densed phase. Stirring problem is the problem of
Bose–Einstain condensation of excitons [1,2] with its
interesting physical consequences: superfluidity, nar-
rowing of optical bands and others. But usually elec-
tron-electron correlation energies in condensed phase
are larger than energy of interaction between excitons
and consequently excitons lose their individuality and
well known and well good studied state of elec-
tron-hole liquid is created [3]. The creation of elec-
tron-hole liquid phase was observed and studied in ger-
manium, silicon and other materials [4–7].
The crystal Cu2O was considered for some time as per-
spective system for exciton Bose–Einstein condensation
because the exciton in Cu2O has small radius and large
value of binding energy. But due to Auger decay pro-
cesses it is difficult to create in this crystal the exciton
density that satisfies Bose–Einstein condencation crite-
rion [8]. So the observation of Bose–Einstain condensa-
tion is not conformed in bulk materials.
Recently intense investigations of excitons under in-
tensive irradiation were carried out for indirect excitons
in quantum wells (QWs) [9–16]. Indirect excitons in
double QW structures with electrons and holes driven to
the separate wells by an electric field are long living par-
ticles. The enhanced lifetime makes it possible to create
large concentrations of excitons and study effects of the
exciton-exciton interaction and a possibility of an exci-
ton condensation, in particular. The authors [9] ob-
served an appearance of narrow line in luminescence
spectra in the crystal with double quantum well at some
threshold value of pumping. In AlGaAs and InGaAs
based structures, a nontrivial feature in the photo-
luminescence spectra of indirect excitons was observed
[11–16]. A spot of the laser excitation was reported to
be surrounded by a concentric bright ring separated
from the laser spot by an annular dark intermediate re-
gion. The distance between the ring and the spot grew
with increasing the intensity of the pumping and rea-
ched up to hundreds �m exceeding by much the exciton
diffusion path. At low temperatures, the external ring
fragmented into a structure with a strongly evaluated
periodicity on a macroscopic scale [11,12,15].
2. Model of system. Basic equations
In the paper a model of exciton condensation is de-
veloped which is based on the traditional phase transi-
tion theory, generalized for the case of a nonequilib-
© V.I. Sugakov, 2006
rium system, in which particles (excitons) have finite
lifetime. The condensed exciton phase is assumed to
exist, and its density is higher than that of the gaseous
phase. In this case, the condensation takes place in
real space, not in momentum space at k � 0, i.e., the
condensed phase is not the Bose–Einstein condensate.
The reasons for the application of the traditional
phase transition theory for the description of the
exciton system are following:
Real QWs comprise defects of different types. Even
at perfect boundaries in semiconductor alloys like
Ga Al As1�x x there are always composition fluctuations
either in the well or in the barrier, or both. The authors
of [17] showed that the disorder sufficiently toughens
the criteria for the appearance of quantum coherence ef-
fects. Disorder leads to broadening of luminescence line.
For the line, which the authors [9,10,15] connect with
the condensed phase, the width is Q � 0.1–1 meV. Ac-
cording to the calculation [17], forQ � 0 5. meV the con-
densed phase can appear only for the densities more than
2 5 1011 2. � �cm . In the case of the smallest disorder ob-
served in experiment (Q � 01. meV) at the exciton den-
sity 5 1010 2� �cm , the Kosterlitz–Thouless transition
happens at T � 0 4. K, which is significantly lower than
the critical temperature in the absence of disorder. The
understanding of the toughening of the quantum criteria
is possible from the following argumentation. The pres-
ence of disorder gives rise to the exciton scattering with
wave vector changing in the well plane, and this causes
broadening of the exciton emission line width Q. The
free path time � can be estimated from the relation
�Q � �. Hence, for the line width Q � 0.1–1 meV the
time of the free path equals to 0.7–7 ps, and for the aver-
age exciton velocity corresponding to T � 1 K the free
pass length v� � 7–70 nm. Therefore, for the exciton
density of 1010–1011 cm–2, the wave function loses its
coherence due to the scattering on defects at the dis-
tances smaller than the distance between excitons. This
makes doubtful the explanation by means of cooperative
quantum effects the phenomena observed at T � 1 K
[11,15]. So, on the one hand, the condition for exciton
collective quantum effects is facilitated in comparison
with the condition for the system of alkaline metal
atoms, in which Bose condensation was observed [18],
because the exciton mass is sufficiently smaller than
atom mass. On the other hand, these conditions become
tougher, because of the presence of random forces due to
such external to the system factors as the spatial disor-
der, the interaction with phonons, etc.
For the formation of the condensed state in real
space, the attractive interaction between excitons is
required. At the same time, owing to the presence of a
dipole moment, directed perpendicularly to the well
plane, repulsive interaction is realized for indirect
excitons at large distances between them. However, at
close distances, the exchange interaction between
electrons and between holes has the essential contribu-
tion. It is known that in a single QW bases on
AlGaAs, a biexciton exists with a relatively large
binding energy (23%) of the exciton binding energy
[19]. The exchange interaction between the electrons
and between holes of two excitons, located in the same
well, takes place in the double QW, too. Therefore,
one has to expect an attraction between excitons lo-
cated at close distances (several exciton radii) at not
very large distances between QWs, when the dipole
moments of excitons are not very large. Thus, the cal-
culations [20,21] of multiexciton system of indirect
excitons taking into consideration exchange energy,
Van-der-Waals energy and correlation energy showed
that the energy of the system as the function of the
density has a minimum at nonzero exciton density for
not very separated quantum wells (d a� 11. , where d is
the distance between the wells, a is the exciton radius
in the well), when the dipole—dipole repulsion is not
too large (see Fig. 2 in the [21]). In this case, the ap-
pearance of excitonic liquid state become possible in a
double QW. The authors also showed that in these
conditions, the excitonic liquid is more advantageous
energy-wise than the formation of biexcitons. Using
the parameters of GaAs crystal and the results of pa-
per [21], it is can be shown that the binding energy
per exciton is negative (i.e., there is an attractive
interaction) at the exciton density of order of
(1.1–2.5)�1011 cm–2. This value is the exciton density
in the condensed phase. In the model considered here,
the system consists of both condensed and gas phase
regions. At critical parameters of system the fraction
of condensed phase islands is small. So, the critical
density is determined by the mean value of the den-
sity, which is of several times smaller than the con-
densed phase density. Therefore, the theory predicts
the realistic value of the density of several units of
1010 cm–2, at which the condensation can occur.
Besides the phase of a dielectric liquid phases with
different polarization can be realized in double QWs.
The phase diagrams of these phases are presented in
[22].
The sizes of condensed phase regions are restricted
both from above and from below. The radius of new
phase nucleus must be larger some threshold which is
determined by a surface energy. A presence of the sur-
face energy determines the shape of condensed phase,
in two-dimensional case the condensed phase has
disk-like shape. A restriction of the size from above
caused by nonequilibrium system which is connected
with exciton creation and exciton decay. In some
range of pumping a number of excitons created by
1450 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
V.I. Sugakov
light in unit time inside of condensed phase region is
less than the number of excitons which disappear. In
this case the steady state of the system is possible if
losses of excitons inside of condensed phase compen-
sates by an exciton inflow from surrounding region of
exciton gas phase. Such compensation is not possible
at large radius of exciton condensed phase. As the re-
sult in the two-dimensional case the condensed phase
must exist as a system of islands similar to elec-
tron-hole drops in bulk semiconductors. The condition
of free energy minimum is satisfied if the islands have
the form of disks. In studied system the islands should
be placed on the ring where the exciton density has
maximum. Since islands of condensed phases takes
excitons from the same source, two islands can not be
situated close one to another. Moreover, the distance
between islands can not be large because in this case
the exciton density between them becomes greater
than a critical value and as the result a new island may
appear. So, there is specific interaction between
condensed phases through the exciton concentration
fields and as a result there is optimal distance between
islands.
To study the creation and structure of the exciton
condensed phase we used the kinetic equations which
describe a nucleation and growth of a new phase. Such
theory was suggested for a description of a liquid
phase formation from supersaturated gas, spinodal de-
composition dynamics [23] and electron-hole drops in
semiconductors [24–27]. There are several original pe-
culiarities of the presented here paper, namely: 1) the
consideration of two-dimensional problem, 2) taking
into account the mutual influence of condensed phase
islands, 3) the investigation of nonuniform particle
distribution, 4) taking into account the finite value
lifetime and the pumping of excitons.
The distribution of exciton density at a steady state
irradiation will be determined from a joint solution of
the system of equations for the size distribution func-
tion of islands and the exciton diffusion outside is-
lands. The size of the islands are determined by four
processes: the creation of excitons by the pumping, the
capture of the excitons from environment, an escape of
excitons from the island, and the exciton decay. Let us
introduce the distribution function fn , which deter-
mines the probability of the island to have n excitons.
The distribution function satisfies the following ki-
netic equation:
�
�
�
�
f
t
j jn
n n1 , (1)
where jn is the probability current for transitions be-
tween island states with n and (n 1) excitons, while
jn�1 stands for transitions between the states with
(n
1) and n excitons,
j W n f W n fn n n� � �
�
( ) ( )( ) ( )1 1, (2)
W ( )� is the probability of transition in unit time with
increasing (decreasing) number of excitons on unity.
The values will be determined later.
We shall study the steady-state solutions of Eq. (1).
There are two types of stationary solutions. In the first
case j j jn n� � ��1 const. In the theory of phase tran-
sitions this solution is used for a determination of a rate
of phase nucleation. In the second case
j jn n� ��1 0. (3)
The solution under the condition (3) describes the dy-
namic equilibrium state which is formed after some
time of pumping action. Such type solution we will
investigate in future. Setting � � �f / tn 0 and jn � 0 in
(2) we obtain the following steady-state solution:
f f
W m
W m
n
m
m n
�
�
�
�
�
�
�
�
�
�
�
�1
1
1
exp ln
( )
( )
( )
( )
. (4)
As we shall see the distribution function has sharp
maximum at some value of n, which is much larger
than unity (n �� 1). Islands with a large number of
excitons and with a shape of disks are described by
radiuses R n/n
/� ( )� 1 2 instead of n. In this case the
probabilities of W n( )( )� è W n( )( )� are satisfy the
following conditions:
W R Rc R W R R Gfi n
( )( ) ( ) ( )� �
2 2� � , (5)
W R Rc W R R c /i if i
( )( ) ( )� �
2 2� � �ex , (6)
where W R W n( ) ( )( ) ( )� �� , Wfi and Wif are the
probabilities for the exciton to be captured by the
disk and to escape from the disk per unit length of the
circle in unit time and per one exciton of disc circle,
respectively, G G r / Rn n n� � ( )dS � 2 is the mean value
of the exciton pumping G r( ) over the nth island area,
c R( ) and ci are the exciton densities on the circle of
the disk and inside the disk.
The relationship between the transition probabili-
tiesWfi andWif may be obtain using the detailed ba-
lance principle. In the case of infinite exciton lifetime
and absence of pumping the equilibrium state between
condensed phases is formed due to processes of exciton
exchange between the island and an environment.
Then in an equilibrium state the following condition
should take place
W R c W cfi if i( ) eq � . (7)
Exciton condensation in quantum wells
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 1451
Where ceq is the density of exciton gas which is in
equilibrium state with the island. The Eq. (7) can be
rewritten in the form
W R
W R
W
W c RkT
if
fi
if
fi i
( )
( )
( )
( )
exp�
�
�
�
�
�
�
��
�0 , (8)
where Wfi ( )� and Wif ( )� are the transition probabil-
ities in the case of a straight line boundary between
condensed and gas phases, W /W c /cfi if i( ) ( )� � � � ,
c� is the equilibrium concentration of excitons on the
straight division line
c c /kT� � 10 exp( )� , (9)
� is the energy condensation per one exciton, c10 �
� � �( )*m kT/2 2
� , m* is the effective exciton mass, � is
the degeneracy of the exciton state, �0 is the surface
strain.
The calculation of these probabilities needs an ap-
plication of quantum—mechanical calculations. For
an estimation of these values we shall calculate the
exciton flux from gas phase in direction of the con-
densed phase. To penetrate in an island exciton should
overcome barrier caused by a dipole—dipole repulsion
between excitons.
Hereinafter we introduce the radius distribution
function f R( ) instead of the particle distribution func-
tion fn by the condition f R f dn/dR Rc fn i n( ) � � 2� .
Let us introduce the dimensionless variables: ~R �
� �R c c R c R /ci
/
i( ) , ~( ~) ( ) ,1 2 � �fi fi i
/W c� ex ( ) ,1 2 ~G �
�G /ci�ex . After some transformations the radius dis-
tribution function (4) may be presented in the form
f R f s Rb( ~) exp( ( ~))� , (10)
where
s R
c R RG
c /R /T
fi
fi
( ~) ln
~( ~) ~ ~
exp( ( ~) ) ~�
2
2
2 10
�
�
� � � R
RdR
R
R
s
�
�
�
�
�
��
~ ~,
~
~
(11)
f f
W m
W m
b
m
ns
�
�
�
�
�
�
�
�
�
�
�1
1
1
exp ln
( )
( )
( )
( )
, (12)
where n s is the value of order of several unit
(n s � 10), n R cs s i� � 2 . The factor fb marked out spe-
cially since the sum in exponent of it can not be re-
placed by integral.
In the stationary case, the equation for the exciton
density c R( ~) in the region outside the islands satisfies
the diffusion equation
D c r
c r
Gex
ex
� 2 ( )
( )
�
�
, (13)
where Dex is the exciton diffusion coefficient.
The solution of the equation (13) should satisfy the
following boundary conditions: the current of excitons
to every island should be equal to the difference be-
tween the number of excitons captured by the island and
the number of excitons which escape from the island. So,
the boundary conditions on ith disk has the form
2 2� �R D c R W c W ci i i fi i iex ( ( ) )| ( ( ( ) )| )� � � r n r if ,
(14)
where Ri is the radius of the ith island, n is the out-
ward and unitary normal vector to the circular
boundary of the ith island, the notation �i means
that the boundary conditions of Eq. (14) are applied
at the surface of an island.
The solution of equation (12) may be written in
the following form:
� �c n r A K /l r mim m i
mi
i( ) ( ) ( ) cos( ( ) ),r r r�
�
�
��ex
0
�
(15)
where
� �n r
D
K /l G dSex
ex
( ) ( ) ( )� � � ��
1
2 0�
r r r , (16)
K xm( ) is the modified Bessel function, ri is the posi-
tion of the center of the ith island, � i r( ) is the polar
angle of the point r in the coordinate system with the
origin at the center of the ith island, l D /� ( )ex ex�
1 2
is the exciton diffusion length. The first term in the
expression (15) is the particular solution of the inho-
mogeneous equation (13). The second term is the
solution of the homogeneous equation (13). So, the ex-
pression (15) satisfies the equation (13). The unknown
coefficients Aim are to be determined from the bound-
ary conditions (14). The solution (15) has distinct
meaning: the first term describes the exciton density
created by pumping in the case of absence of condensa-
tion phases, the second term describes the perturbation
of the exciton density caused by presence of condensed
phase islands which capture excitons.
3. Uniform pumping
In this section we shall investigate an uniform cre-
ation of excitons in double quantum well [28], i.e.,
the pumping does not depend on space coordinates. In
this case the first term in (15) equals
n Gex ex� � . (17)
Firstly we consider the behavior of the system in a
vicinity of threshold of condensed phase formation. In
this case the mutual influence between islands can be
neglected and single island may be studied. The solu-
1452 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
V.I. Sugakov
tion (15) gives together with boundary condition (14)
the following dependence of exciton density outside of
island with a center in the origin of coordinates (in di-
mension units)
� �~(~) ~ ( ~) (~ ~
)c G A R K /lsr r�
00 0 , (18)
where
A R
G c /R /T
K R/l l K
s
00
10
1
( ~)
[~ ~ exp( ( ~) )]
( ~ ~
)
~�
� � �
� 0( ~ ~
)R/l
. (19)
The radius distribution function obtained from (10)
for different pumping is presented in Fig. 1. The
function has sharp maximum for exception of the
pumping very closed to the threshold value. The
maximum determines the most probable meaning of
radius of the island. The threshold value of pumping
can be obtained from two conditions:
� � � � � �f R / R f R / R( ~) ~ ( ~) ~2 2 0 . (20)
We calculated the phase diagram «critical pumping
—temperature» using the formulae (10), (20). Param-
eters (m D*, ,ex ex� ) were chosen for GaAs, �, �, Td
were fitting parameters. Results are presented in
Fig. 2, where points are experimental data from [9]. It
is seen that theory describes the experiments without
suggestion about Bose–Einstein condensation.
Many islands arise with increasing pumping. There
is specific interaction between islands through exciton
concentration fields. In studied system the islands
should be localized on the ring where the exciton den-
sity has maximum. Since islands of condensed phases
takes excitons from the same source, two islands can
not be situated close one to another. Moreover, the
distance between islands can not be large because in
this case the exciton density between them becomes
greater than a critical value and as the result a new is-
land may appear. Thus, there is optimal distance be-
tween islands.
The connection between islands occurs due to the
dependence of the exciton density c R( ~) in (11) at the
surface of the considered island versus the presence of
other islands. We shall make two approximations in
further calculations. We shall suggest that the distance
between islands is larger than the island radius. Under
such condition the exciton density perturbation caused
by the presence of some island slowly changes in limits
of a size of other island and this perturbation may be
considered as constant in the limits of island size. It
means that only terms with m � 0 gives the contribu-
tion in the sum of the expression (15). So, Aim � 0 at
m � 0. Also we suggest that this perturbation is formed
by many islands and as result it may be studied in a
meanfield approximation. Let us consider the exciton
density in vicinity of some island, for example, with
i � 0. In mean field approximation all coefficients Ai0
in the sum of (15) can be replaced by their average
value Ai0 in all terms for exception of the term with
i � 0, the sum over i can be changed by integral in a
suggestion that islands are distributed in space with
density c N/Sisl � , where N is the number of islands, S
is the area of quantum well. Such approximation al-
lows to simplify the boundary condition (14). After
some transformations from boundary conditions (14)
we obtain the following expressions for exciton density
on the surface of the island with the radius ~R:
~( ~) ~( ~,~ ) ~ ( ~, ~,~ ) ( ~ ~
)c R c R c G A R R c K R/l� �
isl isl00 0
2 00
2�A R c l c( ~,~ )
~ ~
isl isl , (21)
where
Exciton condensation in quantum wells
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 1453
5 10 15 20 25 30 35
1
2
3
Radius, m
D
is
tr
ib
u
tio
n
fu
n
ct
io
n
0.5
0.4
0.3
0.2
0.1
0
0.6
Fig. 1. Radius distribution function for single condensed
phase island in quantum well at different pumping ~
G:
0.00715 (1), 0.01 (2), 0.015 (3). � � 10 K, a � 4 K, T � 2 K,
~ ,l ci� � �320 1011 2cm .
0.5
0.4
0.3
0.2
0.1
0.5 1.0 1.5 2.0 2.5 3.0 3.5
T, K
Gc
0
Fig. 2. The dependence of threshold value of pumping on
temperature (solid curve). � �� �3 1K K, , Td � 25. ,K
~ ,l � � �100 10 8�ex s. The points correspond to the experi-
mental data from [9].
A R R c
G A R c l c c
00
00
2
12
( ~, ~,~ )
( ~ ( ~,~ )
~ ~ ~
isl
isl isl�
� � 0 0 0
1 0
exp( ( ~) ))
( ~ ~
)
~
( ~ ~
)
� �
�
/R /T
K R/l l K R/l
, (22)
A R c
G c /R /T
l K R/
00
10
1
( ~,~ )
( ~ ~ exp( ( ~ ) ))
~
( ~ ~isl �
� � �
l K R/l l c) ( ~ ~
)
~ ~
� ��0
22 isl
,
(23)
~R is the mean radius, which coincides with most
probable one.
Let us consider the ensemble of identical systems in
which may be realized the state with different number
of islands. In system with many islands the distribu-
tion function depends on radiuses (or numbers of
excitons) of all islands and determined by the func-
tion f R R R R Ni N( ~ , ~ ~ ~ ; )1 2 � � � �� � . The equation (1)
should be rewritten for many-islands system. In the
model of self-consistent field the distribution function
can be present as product of distribution functions for
separate islands
f R R R R N s R ci N i
i
N
( ~ , ~ ~ ~ ; ) exp ( ~ ,~ )1 2
1
� � � � �
�
�� � isl isl . (24)
After integrating of this value over radiuses of all
islands we obtain the probabilities of a realization of
the state with N islands (or the probability of the
state with the density ~cisl )
F c Sc s R c(~ ) exp( ~ ( ~,~ ))isl isl isl isl� , (25)
s R c
c R RG
c /
fi
fi
isl isl( ~,~ ) ln
~( ~) ~
exp( (
�
2
2
2 10
�
�
� � � ~) ) ~
~ ~
~
R /T R
RdR
R
R
s
�
�
�
�
�
�� . (26)
The most probable density of islands is determined by
the condition
�
�
�
( ( ~,~ ))
~
c s R c
c
isl isl isl
isl
0. (27)
Intensity of the emission, created by excitons of
condensed phase is proportional to the number of
excitons, which islands contain
I R cT � ~ ~2
isl . (28)
In approximation (28) we neglect by the stimulating
emission.
The results of temperature dependence calculation
of emission intensity according (28) are presented in
the Fig. 3. The dependence is described by straight
line. The line crosses the axis at the temperature for
which the considered pumping has threshold value.
Such behavior of intensity coincides with empirical
formula obtained from experiment in the papers [9]
I
T
TT
c
1 . (29)
So, we have explained the experiments without sug-
gestion about Bose–Einstein condensation of excitons.
4. Formation of luminescence from the ring
outside the laser spot
As it was mentioned in Introduction, the authors
of [11–16] observed the emission from the ring outside
the laser spot at the distances from a center of the spot
much larger than exciton diffusion length. At low
temperatures, the ring fragmented into a periodical
structure [11,12,15].
The mechanism of ring appearance was suggested
[15,16] basing on two assumptions: 1) without light
irradiation the well is populated with a certain density
of electrons; 2) holes are captured by the well with a
1454 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
V.I. Sugakov
1.4 1.5 1.6 1.7 1.8 1.9
0.02
0.04
0.06
0.08
TI
2
3
1
0.1
T, K
Fig. 3. The dependence of luminescence intensity on tem-
perature at different pumping, ~
G: 0085. (1), 0.11367 (2),
0.1644 (3). Parameters are the same as in Fig. 2.
larger probability than electrons. As the result, the ir-
radiated structure develops two differently charged
spatial regions. The laser spot and the region around it
are charged positively while the region far away re-
mains negative. Electrons and holes can recombine
only where they meet, therefore, the sharp lumines-
cence ring is formed on the boundary between regions
of opposite charges.
The model of the fragmentation observed in
[11,12,15] was bild in [29,30] on the base of the tradi-
tional phase transition theory, generalized for the case of
a nonequilibrium system, in which particles (excitons)
have finite lifetime. The recombination of electrons and
holes on the ring determines the exciton pumping. Ac-
cording to [15,16] the exciton density has sharp maxi-
mum at the some distance from laser spot. If the density
exceeds the threshold value the islands of condensed
phase are generated on the ring (see insert in Fig. 4).
The system can be described by kinetic equations (1)
with interaction between islands through exciton con-
centration fields. The study of sizes of islands (frag-
ments) and the distance between them is similar to the
consideration of island parameters at uniform pumping
presented above. Results of such study are presented in
[29,30]. The probability for system to have N islands is
described by the formula
f N f S N( ) exp( ( ))� 0isl isl . (30)
The most probable number of islands is determined
by extremum of function S Nisl ( ). The results of taken
from [29,30] calculations of temperature dependence
of island radius, distance between islands and fluctua-
tions of the distances are presented in Fig. 4. The dis-
tance from the laser spot to fragmented ring (r0)
equals 200 �m.
In this section we shall consider the case when the
condensed phase of excitons is shaped as a ring around
the laser spot in region of the largest electron-hole re-
combination. Let the internal and external radii of the
ring formed by the condensed phase are equal to r1 and
r2, respectively, and r0 is the radius of the maximum of
the exciton production rate G r( ) (see the insert in
Fig. 5). The ring grows due to the exciton creation by
pumping and due to the exciton inflow from the envi-
ronment. The ring narrows as a result of the exciton
decay and the exciton escape.
The consideration of the exciton condensation in
the shape of a ring is similar to the study of the forma-
tion of exciton islands considered in the previous sec-
tions. Therefore, we shall set out the problem briefly.
Similarly to the consideration of the islands, let us in-
troduce the distribution function fn , which deter-
mines the probability that the condensed phase shaped
as a ring contains n excitons. The distribution function
obeys to an equation similar to the equation (1), in
which the probability current is equal to
j W r c r r c r fn fi n n n n n�
� � � � �2 1 1 1 1 2 1 2 1 1� ( ( ) ( )), , , ,
� � �2 21 0 1 1 1 2� ��r r G r f W r r c fn n n if n n i n( ) ( )
� �( ), ,r r c f /n n i n2
2
1
2
ex ,
(31)
�where c r n( ),1 and c r n( ),2 are the exciton density on
the internal and the external circular boundaries of
the ring, �r r rn n n� 2 1, , is the thickness of the ring,
G rn( ) is the mean value of the pumping
G r rG r dr/ r rn n
r
r
n
n
( ) ( ) ( )
,
� � �
1
2
0 .
Other designations have been defined in the previ-
ous sections.
Exciton condensation in quantum wells
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 � 1455
d
R
�d/d
14
12
10
8
6
4
2
0
R
,d
,
m
�
1.6 2.0 2.4 2.8
T, K
0.3
0.2
0.1
0
�
d
/d
r0
2 /N�
Fig. 4. Dependences of the island radius R, distance be-
tween islands d and ratio of the island fluctuation shift to
the distance between islands �d/d on temperature. The
insert shows positions of exciton condensed phase islands
around the laser excitation spot.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.5 1.0 1.5 2.0 2.5 3.0
2.4
2.0
1.6
1.2
0.8
/S
S
is
l
r
T, K
r,
m
m
r1
r2
r0
r 1
2
Fig. 5. Dependences of the thickness of condensed phase ring
�r (1) and ratio of values Sisl and Sr (2) on temperature.
The insert shows the condensed phase in the form of a ring.
Further we take into account the fact that n �� 1
and obtain a Fokker–Planck equation for the distribu-
tion function. We assume also that the thickness of
the ring is narrower than the ring radius r rn0 �� � . At
such approximation c r c rn n( ) ( ), ,1 2� .
After reductions similar to those carried out in the
subsection 2, we obtain the following expression for
the distribution function:
f f
W c r W c r c / G r
W cn
fi n if i i n
fi
�
0
1
2
2
2
exp
( ( ) ) ( ( )
(
, � �ex
( ) ) ( ( ),r W c r c / G r
dn
n if i i n
n
1
0
�
�
�
�
�
��� � �ex
. (32)
Coefficients Wfi and Wif are connected by the de-
tailed balance principle of Eq. (8). As the radius of the
ring is significantly larger than the radius of an island
of the previous section we may neglect the surface en-
ergy and put �0 0� .
The exciton density beyond the region of the con-
densed phase obeys Eq. (13). Under condition of the
accepted approximation that the thickness of the ring
is significantly smaller than its radius, the solution of
this equation takes the form
c r n c r/ls( ) exp( )�
ex 0 , (33)
where nex is the solution of the nonuniform Eq.
(13). It describes the exciton density created by the
electron-hole recombination in the absence of the
condensed phase. The value of cs0 is determined by
the boundary condition of Eq. (14). For the ring,
this condition may be written in the form
D
c r
r
W c r W cfi n if iex
�
�
�
( )
( ),1 . (34)
The solution of Eq. (33) with the boundary condi-
tion of Eq. (34), assuming that the thickness of the
ring of the condensed phase is smaller than the diffu-
sion length (�r l� ), reads
c r
n c /T
W l/D
c /Tn
fi
( )
exp( )
exp( ).,1
10
101
�
ex
ex
�
�
(35)
Further we shall use the thickness distribution func-
tion which is defined as f r f dn/d r r c fn i n( )� �� � 2 0� .
Using dimensionless variables, we obtain the following
formula for the thickness distribution function from
Eqs. (32) and (35):
f r f S rr r( ) exp( ( ))� �� 0 , (36)
where
S r
r n c /T r G r /l
r( )
{ [ exp( )] ( ( )(
�
�
�
�
4 2 1 10 10 0� � � �ex )}
[ exp( )] ( ( ))( )
d r
n c /T r G r /l c
�
�
�
�
2 1 1 410 0� � � �ex 10
0
1( ) exp( )
� � �/l /T
r�
. (37)
The most probable thickness of the ring is deter-
mined by the condition
�
�
�
S r
r
r( )�
�
0. (38)
Using Eq. (38), we have calculated the most prob-
able thickness of the ring if the condensed phase is
realized in the shape of a ring at the following values
of parameters: � �� �10 30K K, , ci �
�1011 2cm ,
m m* . ,� 0 37 0 �ex s� �10 7 . The same values of parame-
ters were used in [30] during analysis of a fragment
formation. As seen from Fig. 5, the thickness of the
ring decreases with rising the temperature.
The question to answer is what type of the con-
densed phase is realized? Is it the fragmented ring or
is it the continuous ring? In order to find the solution
of the problem we have performed the following
estimations. We have compared the probabilities of
the appearance of these two states. The curve 2 in
Fig. 5 shows the ratio of the values Sisl and Sr ,
which determine the probabilities according to the
formulas of Eqs. (30) and (36), correspondingly. It
should be noted that absolute values of Sisl and Sr
are large for exception the region of critical tempera-
ture. It means that one of two states (fragmental ring
or continues ring) can been realized with high proba-
bilities depending on the meaning of ratio S /Srisl is
larger or less than unity. It is seen that the state with
the phase distribution in the form of fragments is
more probable than the distribution in the form of
the continuous ring at chosen values of parameters.
In the vicinity to the critical temperature the frag-
mented ring is transformed into continues ring. Such
transformation was observed in experiment [15].
1456 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
V.I. Sugakov
Conclusion
In statistical theory of exciton condensed phase for-
mation in two-dimensional system it is suggested that
condensed phase arises as a result of attractive interac-
tion between excitons. The investigated system is
nonequilibrium one as a result of presence pumping
and finite value of exciton lifetime. It forms some spe-
cific properties such as restricted sizes of condensed
phase regions (islands), a correlation in positions of
islands, so on. The theory was applied to explanation
of experimental manifestation of condensed phase in
quantum wells and also to explanation of the periodi-
cal fragmentation, which was observed in lumines-
cence spectrum from a ring around a laser spot in a
crystal with double quantum wells. For such explana-
tions the theory does not require the exciton
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Exciton condensation in quantum wells
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