Comments: 11 pages, 8 figures Subjects: Mesoscale and Nanoscale Physics (cond-
mat.mes-hall)

# Exciton condensation in quantum wells. Excitonic hydrodynamics. Autosolitons

V.I. Sugakov  Institute for Nuclear Research,National Academy of Sciences
Kyiv,Ukraine<EMAIL_ADDRESS>
###### Abstract

The hydrodynamics equations for indirect excitons in the double quantum wells
are obtained taking into account 1) a possibility of an exciton condensed
phase formation, 2) the presence of a pumping, 3) the finite value of the
exciton lifetime, 4) the exciton scattering by defects. New types of solutions
in the form of bright and dark autsolitons are obtained for the exciton
density. The role of localized and free exciton states is analyzed in a
formation of the emission spectra.

double quantum wells; exciton condensation; solitons

###### pacs:

71.35.Lk, 73.21.Fg

##  I Introduction

While seeking the exciton Bose-Einstain condensation in double quantum wells,
the several non-trivial effects were observed. A very large exciton lifetime
is a special feature of excitons in double quantum well in the presence of an
electric field directed in parallel to the normal of quantum well plain 1 .
The effect occurs due to the separation of electrons and holes into the
different wells, which causes a very weak overlapping their wave function and
the damping of the mutual recombination. The large lifetime allows one to
create a high concentration of the excitons at small pumping and to study a
manifestation of the effects of exciton-exciton interaction. Excitons with
electrons and holes localized in different wells are called ’indirect
excitons’. The nonzero dipole moment of the indirect excitons should cause
their mutual repulsion that complicates the creation of the exciton condensed
phase. These properties and the facts, that excitons have the integer spin and
the small mass, stimulated the search of the exciton Bose-Einstein
condensation in double quantum wells. These investigations gave a number of
new results. So, the narrow band with unusual properties was observed and
studied by Timofeev’s group2 ; 3 ; 4  in the emission spectra of the indirect
excitons in $AlGaAs$ system. It was shown that this band appears at the some
threshold pumping, also the peculiarities of the temperature and pumping
dependencies were revealed. The authors 3 ; 4  builded a phase diagram
”threshold pumping - temperature”.

The nontrivial results were found in a spatial distribution of exciton
emission from quantum well. In the papers 5 , an appearance of a ring outside
the laser spot was observed in the emission from double quantum well. The ring
radius exceeded significantly the exciton diffusion length. The explanation of
the appearance of the ring was given in the papers 7 ; 8  under the
assumption, that holes are captured by the well more effectively than
electrons, and, in addition, there are donors in the crystal, which create
some concentration of free electrons. As a result a region rich by holes
arises in a vicinity of the laser spot. Outside of this region the quantum
well is enriched by free electrons. On the boundary of the region the
processes of a recombination take place, which causes the a creation of
excitons on the ring and the appearance of a spatial distribution of the
emission in the form of the ring.

Intriguing facts appear under an investigation of the spatial distribution of
the exciton density. Different spatial nonhomogeneous structures were observed
in the emission of the indirect excitons at the pumping greater a critical
value. Thus, in the paper 5  a division of the emission ring was observed on
separate fragments periodically localized along the ring. In the paper 9 , in
which the excitation of the quantum well carried out through a window in
metallic electrode, the authors found in the luminescence spectrum a
periodical structure of the islands situated along the ring under perimeter of
the window. The number of the islands growths with increasing of the radius of
the window. The similar results were obtained not only in double quantum
wells, but also in a wide quantum well in the presence of a strong electric
field, which divides electrons and holes between different sides of the
quantum well. As the result the excitons with charges, strong separated in the
space, and with large lifetime are created 10 . On such dipole excitons in the
wide quantum well, the effects similar to those on excitons in double quantum
wells were observed,namely the appearance of periodical structures of the
islands under the window in a electrode11  and on the ring outside the laser
spot All . Recently Timofeev and coauthors 12 ; Gorb  presented examples of
the structures in the emission spectra at the different forms of windows in
the electrode: the rectangle, two circles and others. In the paper 13  the
authors, choosing the form of the electrode, created additional periodical
potential for excitons. It was found, that besides the periodical structure
imposed by external conditions, the partition of the emission into fragments
was observed in the direction, in which the potential is almost uniform.

The phenomenon of the symmetry loss and the creation of structures in the
emission spectra of indirect excitons stimulated a series of theoretical
investigations 14 ; 15 ; 16 ; 17 ; 18 ; 18a ; 18b ; 18d . The authors of the
work 14  considered the instability, which arises under kinetics of level
occupations by the particles with the Bose-Einstein statistics. Namely, the
growth of the occupation of the level with zero moment should stimulate the
transitions of excitons to this level. But the density of excitons was found
greater, and the temperature was found lower than these values observed on the
experiments. Some authors explain the appearance of the periodicity by Bose
condensation of excitons 15 ; 16 . There is a suggestion to describe the
system by a nonlinear Schrodinger equation 17 ; 18 . Also a possibility of the
Mott transition in considered systems was studied 18a . The periodicity
appearance in the exciton system was investigated in 18b  using Bogolyubov’s
equations under some approximations of inter-exciton interaction. Let note the
paper 18d ,in which the observed in 5  appearance of the islands was explained
from classical model diffusion equation with Coulomb interaction between
electrons,holes, and excitons. In the listed above works the main efforts were
applied for the ascertainment of the principal possibility of the appearance
of the periodicity of the exciton density distribution. A specific application
of the results for an explanation of the numerous experiments of different
kind (at different pumping and temperature, nonhomogeneous external fields)
was not employed.

An appearance of an instability of the uniform distribution of the exciton
density and a formation of the periodical distribution were shown in the work
18e  from a position of self-organization processes in non-equilibrium systems
for excitons with attractive interaction. The phenomenon has the threshold
behavior with respect to a pumping. After the successful observation of
periodical structures in the system of indirect excitons in double quantum
well on the base of AlGaAs crystal by Timofeev’s and Butov’s groups, the
several works were fulfilled 19 ; 20 ; 21 ; 22 ; 23 ; 23a ; 23b  in this model
devoted to explanation of the experiments. The theoretical approaches of the
works 19 ; 20 ; 21 ; 22 ; 23 ; 23a ; 23b  are based on the following
assumptions.

1.There is an exciton condensed phase caused by the attractive interaction
between excitons. As was mentioned, there are the dipole-dipole repulsion
interaction between excitons. But the simple calculations show, that the
exchange and van der Waals interactions exceed the dipole-dipole repulsion at
certain distances between excitons, if the distance between quantum wells is
not very far and the exciton dipole moment is not too large. An existence of
attractive interaction between excitons is confirmed by the calculations of
biexcitons 24 ; 25 ; 26 ; 26a , and under investigation of many-exciton system
27 .

2.The finite value of the exciton lifetime plays an important role in the
formation of a spatial distribution of exciton condensed phases. As usually,
the exciton lifetime exceeds significantly the duration of the establishment
of a local equilibrium. By this reason, the lifetime of excitons is suggested
to be equal infinity under the solutions of many exciton problems. But, the
taking into account the finiteness of the exciton lifetime is necessary in the
study of the spatial distribution phases in two-phase systems, because the
exciton lifetime is less than the time of the establishment of the equilibrium
between phases. The last time is determined by slow diffusion processes and is
great. Just the finite exciton lifetime restricts the maximal size of the
exciton condensed phase and it causes the existence of a correlation in
positions of separate regions of the condensed phase. As the result the
spatial structures of the condensed phase appear in the shape of separate
islands (in two dimensional case), parameters of which and the mutual position
depend on the exciton lifetime. Thus, the created spatial structures are non-
equilibrium and they are a consequence of a self-organization in non-
equilibrium systems.

The theory, developed in the works 19 ; 20 ; 21 ; 22 ; 23 ; 23a ; 23b , has
explained many features of the indirect exciton manifestation in the double
quantum wells on the base $AlGaAs$ crystal. So, the observed in 4  behavior of
narrow band as a function of the pumping and the temperature is presented in
20 . It is follow from these calculations in accordance with the experiment 4
, that the intensity of the emission of the narrow band decreases linearly
with temperature at fixed pumping and rises ultralinearly with the growth of
the pumping at fixed temperature . The theory 20  has explained the
experimental ”threshold pumping \- temperature” phase diagram obtained in 3 ;
5 . The works 19 ; 21 ; 22 ; 23  were dedicated to the interpretation of the
appearance of the periodically situated islands in the emission of spectra
from the both the ring outside the laser spot, founded in 5 , and from the
region of the double quantum well under perimeter of the window in electrode,
founded in 9 . The theory described the sizes of the islands, the distances
between them, their appearance and vanishing depending on pumping and
temperature. The calculated behavior of the condensed phase islands around two
laser spots observed in 8  is presented in 23b . At approaching the centers of
lasers spots, the rings of the emission around two spots transform from two
rings around two centers to the deformed single ring with two laser spots
inside the ring in according with experiment 8  . The theory 20  explained an
appearance of a spike observed in But  in emission of the indirect excitons
after the shutdown of a pumping by the increasing of the exciton lifetime in a
consequence of the removal of an Auger processes. In the paper22  the
fragmentation of the inner ring in the laser spot was predicted. These
fragmentation was observed recently in the experiment Rem .

Several theoretical investigations were made for new systems, which were not
studied by experiments so far. It was shown, that a periodical structure of
the condensed phase islands arise at the light irradiation in the quantum well
under electrode with a slot Sug . The chain of the islands moves in the
presence of a linear potential along the slot. The process reminds the Gunn
effect in semiconductors and can be called ”excitonic Gunn effect”. The
pumping dependence study of exciton density distribution in the well under the
electrode in the shape of a disk presents all stages of the phase transitions:
from the islands of the condensed phase in a gas phase till the islands of the
gas phase in an environment of the condensed phase 23c . The investigation of
excitonic pulse moving in external fields 23d  shown that maximum exciton
density remains constant during the exciton lifetime and there is a
possibility of a control of pulse moving by an another laser if the formation
of the pulse occurs by excitons in the condensed phase.

While developing the theory, two approaches of the theory of phase transitions
were used: the model of nucleation (Lifshits-Slyozov) and the model of
spinodal decomposition ( Cahn-Hillart). These models were generalized on the
particles with the finite lifetime, that is important for interpretation of
the experimental results. The involvement of Bose-Einstein condensation for
excitons was not required for the explanation of the experiments, the
considered condensation is the condensation in real space. Among many
experiments, explained by the theory, there are two ones, presented below,
which did not considered jet in the framework of the presented theory.
A. It was shown in the work 30 , that maximum of the frequency dependence of
the emission from the region between the islands is lower, than maximum of the
emission frequency from the islands, so from the region, where the exciton
density is large. The difference of the maxima is small, it is less than the
width of the emission band. But on the base of this date the authors came to
the conclusion, that there is the repulsion interaction between excitons only.
This result contradicts the main assumptions of the model of the works 19 ; 20
; 21 ; 22 ; 23 ; 23a  about the presence of attractive interaction between
excitons, that exist at some distance between excitons and causes the creation
of a condensed phase. In the Appendix we present a possible explanation of
such effect in the case of the attractive interaction between excitons SugUJP
. The explanation is based on the presence in the well of localized exciton
states, levels of which are situated lower than the exciton levels and become
saturated with increasing pumping. The emission band is determined by the free
and localized exciton states. The exciton states form the upper part of the
band. With increasing pumping the number of excitons and the blue part of the
band emission growth. In the case of an attractive interaction, a lowering of
the exciton levels caused by the exciton condensation is small in comparison
with the whole width of the band. As the result the maximum of the emission
shifts to higher side with increasing the pumping, if the emissions from the
exciton condensed phase and from the localized states are not separated.
B. In the works 12 ; 31 ; 32  a coherence was observed in the emission spectra
from island 32 , or even from different islands 12 ; 31 . The coherence was
revealed in the interference of the emission from the different spatial
points.
The effect $B$ is not considered in the presented paper. For its study, the
microscopic model of the condensed phase is needed. But a qualitative
explanation may be given. The interference of the wave functions does not
observed on the experiment directly, the interference of electromagnetic waves
is shown on the experiment. Because the electromagnetic field and scattered
field are coherent, the interference of the emission from two islands may
arise as a result of an imposition of electromagnetic field emitted by some
island and the scattered field by other island. It was shown in the papers 33
; 34 ; 35 , that the strong correlation between exciton densities at different
points takes place in the case when the exciton condensed phase exists. Also
there is a sharp maximum of the Fourier transformation of the two point
correlation function. It is reason of a mutual connection of the wave emitted
from some point with the wave scattered by other region. But the quantitative
calculations require the date of microscopic model of the exciton condensed
phase, particularly, the numerical value of the polarizability is needed.
In the presented paper the hydrodynamics equation for excitons is obtained for
the case, if excitons are in condensed phase. The equations allow to describe
the moving of the complicated system composed of two phases: gas and condensed
ones. We have studied the spatial distribution of both the exciton density and
exciton flux in the case of condensation at steady-state pumping. From
analysis of different solution of the equation for exciton density it was
shown the existence of exciton autosolitons at some parameters of the system.
Also the possible explanation of the effect $A$ is given taking into account
the presence of localized states, which become totaly occupied with increasing
pumping.

It is necessary to emphasize that besides the system, we investigated, there
is an another one, in which the exciton condensation is studied. It is the
exciton condensation in bilayer quantum Hall system. In this system the two
layers are filled by electrons in the presence of strong magnetic field with
total (for two wells) Landau level filling factor $\nu_{T}=1$ and with total
density $n_{T}=n_{1}+n_{2}$, where $n_{i}$ is the density in the separate well
($i=1,2$). An electron in the lowest Landau level of one layer bounded to hole
in the lowest Landau level of the other layer is considered as exciton. The
review of collective effects in such system is presented inEis  . The bilayer
quantum Hall system differs from the system that we consider. The electrons
are created in the bilayer quantum Hall system by donors and the system is
equilibrium. We studied the system in which the excitons are created by the
light,the system is non-equilibrium. The excitons have the finite value of the
lifetime and this fact influences significantly on the behavior of the
collective states, particularly, on the formation of the spatial structures.

##  II Hydrodynamics of exciton condensed phase

The hydrodynamic equations of excitons were obtained and analyzed in the work
36 . In comparison with this paper, we obtain SugUJP  the hydrodynamic
equations of excitons generalizing the Navier-Stokes equations taking into
account the finite exciton lifetime, the pumping of exciton and the existence
of an exciton condensed phase caused by interaction between excitons. The
system is described by the exciton density $n\equiv n(\vec{r},t)$ and by the
velocity of the exciton liquid $\vec{u}\equiv\vec{u}(\vec{r},t)$. The equation
of the continuity of the exciton density is rewritten in the form

$\frac{\partial n}{\partial t}+div(n\vec{u})=G-\frac{n}{\tau_{ex}},$ (1)

where $G$ is the pumping (the number of excitons created for unit time in unit
area of the quantum well), $\tau_{ex}$ is the exciton lifetime. In the
comparison with the typical equation for a liquid, the presented equation for
excitons contains the terms, that describe the pumping and the finite lifetime
of the excitons.

The equation for the movement of the unit volume of an exciton liquid is
rewritten in the form

$\frac{\partial mnu_{i}}{\partial t}=-\frac{\partial\Pi_{ik}}{\partial
x_{k}}-\frac{mnu_{i}}{\tau_{sc}},$ (2)

where $m$ is the exciton mass, $\Pi_{ik}$ is the tensor of the density of the
exciton flux.

$\Pi_{ik}=P_{ik}+mnu_{i}u_{k}-{\sigma}^{\prime}_{ik},$ (3)

where $P_{ik}$ is the pressure tensor, ${\sigma}^{\prime}_{ik}$ is the
viscosity tensor of a tension.

In comparison with the typical Navier-Stokes equation a braking of exciton
liquid by phonons and by defects is introduced in Eq. (2). In the equation (2)
we neglected by the momentum change caused by the creation and the
annihilation of excitons. Indeed, the momentum change in the unit time and in
the unit volume owing to an disappearance of the excitons has the order
$mnu/\tau_{ex}$. Since $\tau_{ex}>>\tau_{sc}$ this value is much less the last
term in the formula (2). The momentum change due to the addition of new
excitons by pumping is small too, because mean exciton moment, created by
external light, is close to zero.

Introducing coefficients of the viscosity and using Eq. (1), Eq.(2) may be
rewritten in the form

$\displaystyle\rho\left({\frac{\partial u_{i}}{\partial
t}+\left({u_{k}\frac{\partial}{\partial
x_{k}}}\right)u_{i}}\right)=-\frac{\partial P_{ik}}{\partial x_{k}}+\eta\Delta
u_{i}$ $\displaystyle+(\varsigma+\eta/3)\left({\frac{\partial}{\partial
x_{i}}}\right)div\vec{u}-\frac{\rho u_{i}}{\tau_{sc}},$ (4)

$\rho=mn$ is the mass of excitons in unit volume.

Let us consider the tensor of pressure. To find the connection between the
tensor and others parameters it is needed to use the equation of the state. We
suggest that the state of the local equilibrium is realized and the state of
the system may be described by a free energy, which depends on spatial
coordinate. Let us present the functional of the free energy in the form

$F=\int{d\vec{r}\left({\frac{K}{2}(\nabla n)^{2}+f(n)}\right)}.$ (5)

The first term in the integrand (5) describes the energy of non-homogeneity.

The pressure tensor will be obtained from equation of the state of the system.
At the given presentation of the free energy, the pressure tensor is
determined by the formula 37

$P_{\alpha\beta}=\left({p-\frac{K}{2}(\vec{\nabla}n)^{2}-Kn\Delta
n}\right)\delta_{\alpha\beta}+K\frac{\partial n}{\partial
x_{\alpha}}\frac{\partial n}{\partial x_{\beta}},$ (6)

where $p=n{f}^{\prime}(n)-f(n)$ is the equation of the state, $p$ is the
isotropic pressure.

Taking into account (6), we rewrite the equation (II) finally in the form
SugUJP

$\displaystyle\frac{\partial u_{i}}{\partial t}+u_{k}\frac{\partial
u_{i}}{\partial x_{k}}+\frac{1}{m}\frac{\partial}{\partial
x_{i}}\left({-K\Delta n+\frac{\partial f}{\partial n}}\right)+\nu\Delta u_{i}$
$\displaystyle+(\varsigma/m+\nu/3)\left({\frac{\partial}{\partial
x_{i}}}\right)div\vec{u}+\frac{u_{i}}{\tau_{sc}}=0.$ (7)

Eqs.(1,II) are the equations of hydrodynamics for an exciton system. These
equations differ from the hydrodynamic equations, investigated in 36 , by the
presence of the terms of the right side in Eq.(1), which take into account the
lifetime and pumping, and by the presence of the third term in Eq.(II), which
describes a condensed phase. It follows from the estimations, made in the work
36 , that the terms with the viscosity coefficients are small and we shall
neglect them.

In the case of a steady state irradiation of the system, the Eqs.(1) and (II)
have the solution $n=G\tau$, $u=0$. In order to investigate the stability of
this solution we consider the behavior of a small fluctuation of the exciton
density and the velocity from these values: $n\rightarrow n+\delta
n\exp(i\vec{k}\cdot\vec{r}+\lambda t)$, $u=\delta
u\exp(i\vec{k}\cdot\vec{r}+\lambda(\vec{k})t)$. After substitution these
expressions in Eqs.(1, II), we obtain in the linear approximation with respect
to the fluctuations the following expression

$\displaystyle\lambda_{\pm}(\vec{k})=\frac{1}{2}(-(1/\tau_{sc}+1/\tau_{ex})$
$\displaystyle\pm\sqrt{(1/\tau_{sc}-1/\tau_{ex})^{2}-\frac{4k^{2}n}{m}(k^{2}K+\frac{\partial^{2}f}{\partial
n^{2}})}),$ (8)

It is follow from (II), that both parameters $\lambda_{\pm}(\vec{k})$ have a
negative real part at small and large values of vector $\vec{k}$ and,
therefore, the uniform solution of hydrodynamics equation is stable. But the
value $\lambda_{+}(\vec{k})$ may be positive in some interval of vector
$\vec{k}$ , when $\frac{\partial^{2}f}{\partial n^{2}}$ becomes negative. In
these case the uniform distribution of the exciton density is unstable with
respect to a formation of nonhomogeneous structures. The instability arises at
some threshold value of exciton density and at some critical value of the wave
vector. Analysis of the equation (II) gives the following expression for the
critical values of the wave vector $k_{c}$ and the exciton density $n_{c}$

$k_{c}^{4}=\frac{m}{Kn_{c}\tau_{sc}\tau_{ex}},$ (9)
$\frac{k_{c}^{2}n_{c}}{m}\left(k_{c}^{2}K+\frac{\partial f(n_{c})}{\partial
n_{c}^{2}}\right)+\frac{1}{\tau_{sc}\tau_{ex}}=0.$ (10)

For stable particles ($\tau_{ex}\rightarrow\infty$) the equations (9,10) give
the condition $\frac{\partial f}{\partial n^{2}}=0$, that is condition for
spinodal decomposition for a system in the equilibrium case.

Depending on parameters the Eqs.(1,II) describe the ballistic and diffusion
movement of the exciton system. The relaxation time $\tau_{sc}$ plays the
important role in a formation of the exciton moving. Due to arising of the
nonhomogeneous structures, the exciton currents in the system exist
($\vec{j}=n\vec{u}\neq 0$) even under the uniform steady-state pumping.
Excitons are moving from regions with the small exciton density to the regions
with the high density. In the presented paper we shall consider the spatial
distribution of exciton density and exciton current in the double quantum well
under steady-state pumping. In this case the exciton carrent is small and we
suggest the existence of the next conditions

$\frac{\partial u_{i}}{\partial t}<<u_{i}/\tau_{sc}.$ (11)
$u_{k}\frac{\partial u_{i}}{\partial x_{k}}<<u_{i}/\tau_{sc}.$ (12)

Particularly, the Eq.(11) holds under the study of the steady-state exciton
distribution. The fulfilment of Eq.(12) will be shown later after some
numerical calculations.

Using the conditions (11) and (12) we obtain from Eq.(II) the value of the
velocity $\vec{u}$

$\vec{u}=-\frac{\tau_{sc}}{m}\vec{\nabla}\left({-K\Delta n+\frac{\partial
f}{\partial n}}\right),$ (13)

As the result the equation for the exciton density current may be presented in
the form

$\vec{j}=n\vec{u}=-M\nabla\mu,$ (14)

where $\mu=\delta F/\delta n$ is the chemical potential of the system,
$M=nD/\kappa T$ is the mobility, $D=\kappa T\tau_{sc}/m$ is the diffusion
coefficient of excitons.

Therefore, the equation for the exciton density (1) equals

$\displaystyle\frac{\partial n}{\partial t}=\frac{D}{\kappa
T}(-Kn\Delta^{2}n-K\vec{\nabla}n\cdot\vec{\nabla}\Delta n)$
$\displaystyle+\frac{D}{\kappa
T}\vec{\nabla}\cdot\left(n\frac{\partial^{2}f}{\partial
n^{2}}\vec{\nabla}n\right)+G-\frac{n}{\tau_{ex}}.$ (15)

Just in the form of (II) we investigated a spatial distribution of the exciton
density at exciton condensation at different dependencies $f$ on $n$ [22, 23,
33, 38, 39]. So, our previous consideration of the problem corresponds to the
diffusion movement of hydrodynamics equations (1,II). At some conditions,
applied to the functional $F$ the uniform solution is unstable, and the
spatial structure arises in the system. For the system under study the
condensed phase appears, if the function $f(n)$ describes a phase transition.
In the papers mentioned above the examples of such dependencies were given.
Here, we analyze an other dependence $f(n)$, which often is also used in the
theory of phase transitions. We shall approximate the density of the free
energy in the form

$f=\kappa
Tn(\ln(n/n_{a})-1)+a\frac{n^{2}}{2}+b\frac{n^{4}}{4}+c\frac{n^{6}}{6},$ (16)

where $a$, $b$, $c$ are the constant values. Three last terms in the formula
(16) are the main terms, they arise due to an exciton-exciton interaction and
describe the phase transition. The first term was introduced in order to
describe the system in a space, where the exciton concentration is small(if
such region exists in the system). With increasing the exciton density the
term $a\frac{n^{2}}{2}$ must manifests itself firstly. It gives the
contribution the $an$ value to chemical potential. The origin of this term is
connected in our system with the dipole-dipole interaction that should become
apparent at the beginning due to its long-range nature. For estimations of $a$
we may use for the dipole-dipole exciton interaction in double quantum well
the plate capacitor formula $an=4\pi e^{2}dn/\epsilon$, where $d$ is the
distance between wells, $\epsilon$ is the dielectric constant. This formula is
used usually for a determination of the exciton density from the the
experimental meaning of the blue shift of the frequency of the exciton
emission with the rise of the density. It is follow from the formula that
$a=4\pi e^{2}d/\epsilon$. This expression is approximate because it does not
take into account the exciton-exciton correlations 25 ; 37b . When the exciton
density growths the last two terms in (14) begin to play a role. An existence
of the condensed phase requires that the value $b$ was negative($b<0$). For
stability of the system at large $n$ the parameter $c$ should be positive
($c>0$). It is suggested in the model, that the condensed phase arises due to
the exchange and van der Waals interactions. For the system with the large
distance between wells the dipole-dipole interaction exceeds attractive
interaction and the condensed phase does not arise. The disappearance of
chemical potential minimum as a function of $n$ with increasing of the
parameter $a$ (with increasing the distance between the quantum wells) is
presented in Fig1.

Figure 1:  Qualitative dependence of the chemical potential on $n$. The
parameter $a$ for the thin line is greater than for the thick line. The
parameters $b$ and $c$ are the same for both curves.

Let us introduce the dimensionless parameters: $\tilde{n}=n/n_{o},$ where
$n_{o}=\left({a/c}\right)^{1/4}$, $\tilde{b}=b/(ac)^{1/2}$,
$\tilde{\vec{r}}=\vec{r}/\xi$, where $\xi=\left({K/a}\right)^{1/2}$ is the
coherence length , $\tilde{t}=t/t_{0}$, where $t_{0}=\frac{\kappa
TK}{Dn_{o}a^{2}}$, $D_{1}=\frac{\kappa T}{an_{o}}$, $\tilde{G}=Gt_{0}/n_{0}$,
$\tilde{\tau}_{ex}=\tau/t_{0}$. As the result the equation (II) is reduced to
the form (hereinafter the symbol $\sim$ will be omitted in the equation)

$\displaystyle\frac{\partial n}{\partial t}=D_{1}\Delta n-n\Delta^{2}n+n\Delta
n(1+3bn^{2}+5n^{4})$ $\displaystyle-\vec{\nabla}n\cdot\vec{\nabla}\Delta
n+(\vec{\nabla}n)^{2}(1+9bn^{2}+25n^{4})+G-\frac{n}{\tau_{ex}}.$ (17)

The solutions of the equation (II) are presented in Fig.2 in the one-
dimensional case ($n(\vec{r},t)\equiv n(z,t))$) for three values of the
steady-state uniform pumping.

Figure 2:  The spatial dependence of the exciton density at the different
value of the pumping: for the continues line $G=0.0055$, for the periodical
line $G=0.008$, for the dashed line $G=0.0092$. $D_{1}=0.03$, $b=-1.9$

The solutions are obtained at the initial conditions $n(z,0)=0$ and the
boundary conditions
$n^{\prime}(0,t)=n^{\prime}(L,t)=n^{\prime\prime}(0,t)=n^{\prime\prime}(L,t)=0$,
where $L$ is the size of the system. The periodical solution exists in the
some interval of the pumping $G_{c1}<G<G_{c2}$. At the great size of the
system the structure of the solution (the period and the amplitude of the
lattice) does not depend on the boundary conditions. At the given parameters
the periodical solution exists at $0.0055<G<0.0092$. Outside this region, the
solution describes a uniform system: the gas phase at the pumping less the
lower boundary value and the condensed phase at the pumping greater upper
boundary value. Upper part of the periodical distribution corresponds to
condensed phase, the lower one corresponds to the gas phase. The size of the
condensed phase increases with the change of the pumping from $G_{c1}$ to
$G_{c2}$. At $G>G_{c2}$ the state with uniform distribution of condensed phase
emerges.

Figure 3:  The spatial dependence of the exciton current at $G=0.008$,
$D_{1}=0.03$, $b=-1.9$.

Fig.3 shows the spatial dependence of the exciton current calculated by the
formula (14). The current equals zero in the centers of the condensed and gas
phases and it has a maximum in the region of a transition from the condensed
phase to the gas phase.

Let us do some estimations. It is seen from Fig.2, that the exciton density in
the condensed phase redouble approximately in comparison with the density in
the gas phase. The results for the currents in Fig.3 is presented in
dimensionless units: $\tilde{j}=j/j_{o}$, where $j_{o}=n_{o}u_{o}$,
$u_{o}=(\tau_{sc}n_{o}a)/(m\xi)$ is the unit of the velocity. The exciton
density presented in Fig.2 in dimensionless units ($\tilde{n}=n/n_{o}$). It is
seen from Fig.2 that$\tilde{n}\sim 1$, and the magnitude of $n$ has an order
on $n_{o}$. So, for estimations we may suppose that $n_{o}a$ corresponds to
the shift of the luminescence line with increasing the exciton density, also
the magnitude of $\xi$ has the order of the size of the condensed phase. For
the following magnitudes of parameters
$\tau_{sc}=10^{-11}\textmd{s},n_{o}a=2\cdot 10^{-3}\textmd{eV},m=2\cdot
10^{-28}\textmd{g},\xi=2\cdot 10^{-4}\textmd{cm}$, we obtain $u_{o}\sim
10^{6}\textmd{cm/s}$. According to calculations (see Fig.3) the magnitudes of
the current and the velocity are in two orders of the values less than their
unit. So, at chosen parameters, the maximal amplitude of the velocity of the
exciton density flux in the dissipative structure in the double quantum well
equals $10^{4}\textmd{cm/s}$. In order to verify the fulfilment of the
condition (12), let us suppose that ($\partial u_{i})/(\partial x_{k}\sim
u/l$), where $l$ is the period of the structure. It follows from experiments 5
; 9  that $l\sim(5\div 10)\mu\textmd{m})$. Using these date we obtain that the
condition (12) is satisfied very good. This condition is violated at
$\tau_{sc}\geq 10^{-9}\textmd{s}$. It is the very large value. The calculation
using uncertainty principle from the band width of the narrow line ($2\cdot
10^{-4}\textmd{eV}$)2 ; 3  gives more less the magnitude $3\cdot
10^{-12}\textmd{s}$. Therefore, the formation of nonuniform exciton
dissipative structures in the double quantum well occurs by the diffusion
movement of excitons. For a proof of the main equation (II) the last term in
Eq. (2), which describes the loss of the momentum due to scattering on defects
and phonons, is of importance. Just this term describes the processes, which
cause a decay of the exciton flux. From the viewpoint of the possibility of
the appearance of superfluidity, the situation for excitons is more
complicated than that for the liquid helium and for the atoms of alkali metals
at ultralow temperatures. In the last systems, the phonons (moving of
particles) are intrinsic compound part of the system spectrum, the interaction
between phonons (particles) are the interaction between of atoms of the system
and does not cause the change of the complete momentum of the system and its
moving as whole. Phonons and defects for excitons are external subsystems,
which brake the exciton moving. Therefore, for the creation of exciton
superfluidity it is needed, that the value of $\tau_{sc}$ growths
significantly. It is possible for exciton polaritons, which weekly interact
with phonons; and there is a certain experimental evidence on an observation
of the polariton condensation 40 . For the indirect excitons the critical
temperature of the superfluid transition is strongly lowered by
inhomogeneities 40a ; 40b . So, the question about the possibility of the
superfluidity existence for the indirect excitons on the base AlGaAs system is
open.

Thus, the peculiarities, that are observed at large densities of the indirect
excitons, may be explained by the phase transitions in the system of the
particles with attractive interactions and the finite value of the lifetime
without an involvement of the Bose-Einstein condensation.

##  III Excitonic autosolitons

As it was shown, at $n<n_{c1}(G<G_{c1})$ the uniform solution of Eq. (II) is
stable. But, at some limits of a pumping at $G<G_{c1}$ there exists a
stationary solution for the exciton density distribution localized in a space.
For example, with the parameters used for calculations of the exciton
distribution in Fig.3, the threshold value of the pumping equals
$G_{c1}=$0.0055; but, at a steady-state pumping there is the spatial
nonuniform solution of the equation (II) at $G<G_{c1}$ in the form of an
isolated spike. It may be obtained solving Eq.(II) at the pumping, which
consist of a constant value $G_{0}$ and an additional pulse $dG$ with the
maxima in the some point of the space and in the time moment

$dG=s\,\exp[-w(z-z_{0})^{2}]\,\exp[-p(t-t_{0})^{2}]$ (18)

where $s,\,w,\,p$ are parameters. The formula (18) describes a pulse of the
pumping, which acts during some time interval with the maximum in the point
$z_{0}$.

The solution of Eq.(II) obtained under an application of the addition pulse
(18) in the region $z_{0}=L/2$ has at $t\to\infty$ the form presented in
Fig.4.

Figure 4:  The spatial dependence of the exciton density of the excitonic
autosoliton at the pumping $G=0.005<G_{c1}$; $D_{1}=0.03$, $b=-1.9$.

The solution exists at $t\to\infty$, i.e. at the times, when the action of the
addition pulse is absent already. The shape of the spike $n(z)$ does not
depend on parameters $s,\;w,\;p$, except cases, when at least one of these
parameters tends to zero and becomes less some value. In addition, the
solution in the form, presented in Fig.4, arises also, if the additional pulse
is absent, but there is some distribution of the exciton density in the
initial time $t=0$:

$n(z,0)=s_{0}\exp(-w(z-z_{0})^{2}).$ (19)

Fig.5 shows the distribution of exciton current in the vicinity of the
localized solution. The current changes the sign in the center of the
localized state of the density.

Figure 5:  The spatial dependence of the exciton current in the localized
state at the pumping $G=0.005<G_{c1}$; $D_{1}=0.03$, $b=-1.9$.

We have verified by direct calculations, that the solution presented in Fig.4
in the form of a localized distribution of the density is stable. We call the
state, that describes this solution localized in space, by excitonic
autosoliton. The spatial dependence of the exciton density will be designated
by $n_{as}(z)$. The autosolitons exist in the some regions of the pumping
$G_{cas}<G<G_{c1}$. At chosen parameters in Fig.4 the autosolitons arise under
conditions $0.003<G<0.0055$. The solutions exist in the form of the
autosoliton side by side with the uniform solutions. The lower boundary
$G_{cas}$ depends on exciton parameters, particulary, on the exciton lifetime.

Excitonic autosolitons correspond to solitary solutions of nonlinear equations
for the excitons (II). The name“auto” is introduced according to 42 ; it
underlines, that the solitary solutions arise in the dissipative system in a
contrast to ”solitons”, which appear in conservative systems. The obtained
here autosolitons correspond to the ”static autosolitons” according to the
classification 42 . The solutions in the form of the autosolitons are
degenerated: if there is a solitary solution $n_{as}(z)$,then
$n_{as}(z-z_{0})$ will be also solution at the arbitrary $z_{0}$ (in the
infinite medium). But, if there is an external field in a system, which
creates a spatial dependent additional potential for excitons, the solitary
excitation moves. Thus, at linear spatial dependence of the additional
potential energy in the formula of the free energy (16) the term $\delta
V=-dz$ should be added. In this case the equation (II) has the solution in the
form of autowaves $n_{as}(z-vt)$, where $v$ is the velocity of the autowave.
In the region, in which the periodic solution of the exciton density takes
place ($G(c1)<G<G(c2)$), such autowaves were investigated in the work Sug .

Localized solutions exist also in the some region at the pumping greater the
value, at which the periodical structure arises ($G>G(c2)$). The dependence of
the exciton density may be obtained from Eq.(II) choosing an additional
pumping pulse in the form (18), but at $a<0$. An example of such solution is
presented in Fig. 6. These structures appear in the form of a dip, and can be
called ”dark autosolitons” by analogy with the soliton’s terminology.

Figure 6:  The spatial dependence of the exciton density of the dark
autosoliton at $G=0.0095>G_{c2}$; $D_{1}=0.03$, $b=-1.9$ .

To explain the appearance of the autosoliton-type solution we recall that the
phase transition are investigated in the paper. As is known, there exists the
region between spinodal and binodal, in which the creation of a nucleus of new
phase is needed for phase transition. The size of the nucleus should exceed
some critical value. The obtained critical value of $n_{c}$ (Eqs.(9,10)) is
based under the consideration of the small fluctuations and corresponds really
to the boundary of the spinodal region corrected by non-equilibrium state of
the system. The autosoliton arises at pulses larger some critical value. So,
the appearance of autosolitons corresponds to the appearance of the nucleus
outside the spinodal boundary for phase transitions of the stable particles. A
size of the new phase of the stable particles increases with time, while the
distribution density of the unstable particles (excitons) does not depend on
the time (at a steady-state pumping). It explains the existence of the
localized states of exciton distribution outside the region of the instability
of an uniform distribution. It should be noted, that in the phase transition
approach of nucleation-growth with an generation on unstable particles 19 ; 20
; 21  the existence of the condensed phase islands outside the spinodal region
is taking into account automatically, because this approach consider
fluctuations, which do not presented in the equation (II).

The model of the nucleation-growth, which we used under the investigation of
the exciton condensation in 19 ; 20 ; 21 , is close to the methods, used in 43
; 44 ; 45  under the investigation of electron-hole droplets in germanium and
silicon. But in contrast to the works 43 ; 44 ; 45  we took into account the
correlation in the spatial positions of the droplets (the islands in the two-
dimensional case), that allows to describe the different structures, mutual
positions of of islands, their concentration, the formation of the periodicity
and other properties, which are observed in quantum wells on the base of the
AlGaAs system.

##  IV Conclusions

In the work, the several problems of the theory of indirect exciton
condensation in quantum well on the base AlGaAs crystal are considered.

The hydrodynamic equations for excitons were build with taking into account a
possibility of the condensed phase formation, the finite lifetime of excitons,
the scattering of excitons by defects. The analysis of the equations was
fulfilled for the diffusive exciton flow. The equations explain many spatial
excitonic structures obtained under experimental investigations of indirect
excitons in quantum wells on the base AlGaAs without an involvement of the
Bose-Einstein condensation of excitons.

Also, solutions of the equations in a form of solitary localized states (the
bright and dark exciton autosolitons) were found. They exist side by side with
the steady-state exciton density state and may be excited by the additional
pulse greater some threshold value. In the model of the nucleation-growth
these solutions correspond to the nuclei of the condensed phase in a gas phase
(the bright autosolitons) and to the nuclei of the gas phase in the condensed
phase (the dark autosolitons).

It was shown, that the results of the experiment 30 , in which has been
obtained, that the emission spectrum from the condensed phase region is
shifted to shortwave side in a comparison of the emission spectra from the
region of the gas phase, do not contradict to the model with an attractive
interaction between excitons. For the explanation it was taking into account,
that a formation of the emission spectrum occurs by both free and trapped
excitons.

##  Appendix A Distribution of excitons between localized and delocalized
states

According to 30  the frequency of the emission from the islands on the ring
around the laser spot is higher, than the frequency of emission from the
region between islands. The authors made the conclusion 30 , that interaction
between excitons is repulsive, and, therefore, the formation of the condensed
phase by attractive interaction is impossible. It contradicts the main
assumption of the works 19 ; 20 ; 21 ; 22 ; 23 , though these works explain
many experiments. Now we remove this contradiction, taking into account the
presence of localized excitons.

Residual donors and acceptors, defects, inhomogeneous thickness of the wells
create an accidental fluctuating potential, which may be the reason of
appearance of the localized levels. Till now the explanation of the creation
of the localized states is not determined definitely, but their existence is
confirmed by the presence of an emission in the region of the frequencies less
the frequency of the exciton band emission and broadening of exciton lines. At
the low temperature and at the small pumping the main part of the band
consists of the emission from defect centers, the part of the exciton emission
growths with increasing pumping. Let us consider the relation between the
contribution to the emission band intensity from free excitons and the
excitons localized on defects. Since the defect structure of the samples
depends on their preparation, a solution of this problem can not be solved in
general. We shall use some approximations.

We shall consider the energy distribution of electron-hole pairs at steady-
state irradiation. Such pair may be the delocalized exciton in the exciton
band and an electron and hole localized near a defect or in the region with a
modified thickness of the well. For small density of the excitation (the
electron-hole pairs) the interaction between them may be neglected. Due to
long-range character the dipole-dipole interaction, which appear with
increasing the excitation density, gives an identical shift of delocalized and
localized levels. It means that this interaction do not influence on mutual
distribution of the localized and delocalized states. We shall suggest that
the localized states are saturable, namely, every center may capture a
restricted number of excitations. In our calculations we shall assume that
only single excitation may be localized on the defect. Another excitations are
or absent or have very low binding energy and are unstable. The dependence of
a density of localized states on energy was chosen in the exponential form,
namely $\rho(E)=\alpha N_{l}exp(\alpha E)$, where $N_{l}$ is the density of
the defect centers, $E$ is the depth of the trap level. The exciton states
(free and localized) are distributed on levels after a creation of electrons
and holes by an external irradiation and their subsequent recombination and
relaxation. Because the time of the relaxation is much less than the exciton
lifetime, the distribution of excitation between free and localized states
corresponds to to a state of thermodynamical equilibrium. In considered model
we should obtain the distribution of electron-hole pairs, a population of
which on a single level may be changed from zero to infinity for $E>0$ (the
free exciton states) and from zero to one for $E<0$ (the localized states).
Formally, in considered system the free excitons have Bose-Einstein statistic
and localized excitations obey Fermi-Dirac statistic. At small exciton density
Bose-Einstein and Boltsmann statistics give the similar results for the free
excitons, but the application of Fermi-Dirac statistic for localized states
with single level for one trap is important. The equation for the energy
distribution may be find from minima of large canonical distribution

$w(n_{k},n_{i})=\exp\left(\frac{\Omega+N\mu-E}{\kappa T}\right),$ (20)

where $N=\Sigma_{i}n_{i}+\Sigma_{k}n_{k}$,
$E=\Sigma_{i}n_{i}E_{i}+\Sigma_{k,l}n_{k}E_{k}$, $n_{i}=0,1$,
$n_{k}=0,1,...\infty$, $k$ is the wave vector of the exciton, $l$ designates
the singular levels. $\mu$ is the exciton chemical potential.

The distribution of excitons over free and localized level is determined from
minimum of the functional (20). As the result we obtain the following
conditions for the mean values of the free exciton density $n$ and the density
of the localized states $n_{L}$

$n_{ex}=\frac{g\nu}{4\pi
E_{ex}a_{ex}^{2}}\int_{0}^{\infty}\frac{dE}{\exp(\frac{E-\mu}{\kappa T})-1},$
(21) $n_{L}=\alpha N_{l}\int_{-\infty}^{0}\frac{\exp(\alpha
E)dE}{\exp(\frac{E-\mu}{\kappa T})+1},$ (22)

where $a_{ex}=(\hbar^{2}\varepsilon)/(\mu_{ex}e^{2})$ and
$E_{ex}=(\mu_{ex}e^{4})/(2\varepsilon^{2}\hbar^{2})$ are the radius and the
energy of the exciton in the ground state in bulk material, $g=4$, $\mu_{ex}$
is the reduced mass of the exciton, $\nu$ is the ratio of the reduced and the
total mass of the exciton. The chemical potential $\mu$ is determined from
condition

$n_{L}+n=G\tau_{ex},$ (23)

where $G\tau_{ex}$ is the whole number of excitation (free and localized) per
unit surface.

The dependence of the distribution of the free and the localized exciton on
pumping is presented in Fig.7 as function of whole number of the excitation
presented in units of $1/a_{ex}^{2}$

Figure 7:  The dependence of the density of the free (thick line) and trapped
(thin line) excitons on the pumping. The parameters of the system:
$T=2K,N_{l}=0.001/a_{ex}^{2},\alpha=300(\textmd{eV})^{-1}$

Let us the exciton radius equals $10\textmd{nm}$. Then the concentration of
the traps and the width of the distribution of trap levels, chosen under
calculations of Fig.7, have the order of $10^{9}\textmd{cm}^{-2}$ and
$0.003\textmd{eV}$, correspondingly.

As it is seen from Fig.7 that the number of the localized excitations exceeds
at small pumping the number of free excitons and the emission band should be
determined by the emission from the traps. With increasing pumping the
occupation of the trap levels become saturated. For chosen parameters the
concentration of excitations at the saturation is a value of the order of
$10^{9}\textmd{cm}^{-2}$. Simultaneously with the saturation of the localized
levels the exciton density growths. As the result, the shortwave part of the
emission band should be increased with increasing pumping. When the exciton
density becomes great, the collective exciton effects begin to manifest
themselves. The equations (21,22) do not take into account the interactions
between the excitations, and special models and theories are needed for
descriptions of collective effects. The appearance of a narrow line was
observed in 2  with increasing pumping on the shortwave part on the exciton
emission band. Simultaneously, the patterns arise in the emission spectra. The
narrow line appeared after the localized states becomes occupied. According to
2  this line is connected with the exciton Bose-Einstain condensation.
According to our model 19 , the islands of condensed phase arise, if the
exciton density become higher some threshold value, and the narrow band
corresponds to the condensed phase, caused by the attractive interaction
between excitons.

Figure 8:  The distribution of excitations in the traps and in the states of
exciton band. The thick line in fig.8a corresponds to the energy per single
exciton in the condensed phase. On the right (fig.8b) the upper line describes
the whole emission from the island (the emission of both the condemned phase
and trapped excitons), the low line describes the emission of the trapped
excitons.

The energy per single exciton in the condensed phase is less than the energy
of free excitons (the thick line in Fig.8), but the gain of energy under
condensation is less than the whole bandwidth, which are formed by the
localized and delocalized states. The gain of energy is significantly less
than the binding energy of the exciton to an electron -hole drop in silicon
and germanium. According to 3 ; 4  the narrow band is shifted to the red side
with increasing pumping in the value less than 0.5 meV, while the whole
bandwidth has the order of 2 meV. So, the energy of photons emitted from the
islands of condensed phase is higher than the energy of photons emitted by
traps. The excitons can not leave the condensed phase (the islands) and move
to the traps (to the states with lower energy) since the levels of the traps
are occupied already. Thus, the emission frequency of the condensed phase is
larger, than the emission frequency of the gas phase, even at the attractive
interaction between excitons. It may be reason of obtained in 30  results in
which the maximum of the frequency of the emission from the islands is higher
than the maximum frequency from the regions between the islands.

The qualitative results coincide with the results obtained from the solution
of kinetics equations in SugUJP  for level distributions. The similar behavior
of the distribution of free and trapped excitons takes place for an other
energy dependence of the density of localized states. We confirmed the results
for the gaussian density distribution.

The results may be applied for explanation of intensity and temperature
dependence of the exciton emission of dipolar excitons in InGaAs coupled
double quantum wells Schin . The authors observed with increasing pumping the
growth of the shortwave side of the emission band, the narrowing of the band.
They obtained that the shortwave side of the band is very sharp. These results
may be explained by suggestion, that the lower part of the band is formed by
localized states. After the saturation of the localized states with increasing
pumping, the excitations begin to occupy the levels of free excitons. Since
the density of exciton state is much greater than the density of defect states
the shortwave edge of the band is sharp. The exciton condensed phase may not
arise in the system investigated in Schin  as the distance between wells
($17\textmd{nm}$) is greater than the distance in the system investigated by
the Timofeev’s group 2 ($13\textmd{nm}$), so the repulsive dipole-dipole
interaction between excitons in Schin  may be larger than the attractive
interaction.

It should be noted, that in the work 5  the band of the emission from the
condensed phase is wider than the narrow band that was observed in the work 4
, where the another method of the creation of excitons was applied. Maybe, it
is related to the fact, that in the work 5 , excitons were created in the
region of the p-n transition: the electrons approach this region from one
side, and the holes move to the region from other side. The excitons are
situated in a space between the regions rich by electrons from one side, and
by holes from other side. To combine into excitons, electrons and holes should
surely go through the region of the condensed phase. Therefore, this region
contains the charges, which create an electric field and cause the broadening
of the emission band.

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²2

11institutetext: Department of Diagnostic and Interventional Neuroradiology,
Klinikum rechts der Isar, School of Medicine, Technical University of Munich,
Germany 22institutetext: icometrix, Leuven, Belgium 33institutetext:
Department of Informatics, Technical University of Munich, Munich, Germany
44institutetext: Department of Diagnostic and Interventional Neuroradiology,
University Medical Center Hamburg-Eppendorf, Germany 55institutetext:
Institute of Diagnostic and Interventional Neuroradiology, University of Bern,
Bern, Switzerland 66institutetext: ARTORG Center for Biomedical Engineering
Research, Univ. of Bern, Switzerland 77institutetext: Chan Division of
Occupational Science and Occupational Therapy, University of Southern
California, Los Angeles, CA, USA 88institutetext: TranslaTUM – Central
Institute for Translational Cancer Research, Technical University of Munich,
Munich, Germany 99institutetext: Medical Imaging Research Center (MIRC), KU
Leuven, Leuven, Belgium 1010institutetext: Medical Image Computing (MIC),
ESAT-PSI, Department of Electrical Engineering, KU Leuven, Leuven, Belgium
1111institutetext: Department of Quantitative Biomedicine, University of
Zurich, Zurich, Switzerland 1212institutetext: Department of Neurology,
Klinikum rechts der Isar, School of Medicine, Technical University of Munich,
Germany

# ISLES 2022: A multi-center magnetic resonance imaging stroke lesion
segmentation dataset

Moritz Roman Hernandez Petzsche 11 Ezequiel de la Rosa 2233 Uta Hanning 44
Roland Wiest 55 Waldo Enrique Valenzuela Pinilla 55 Mauricio Reyes 66 Maria
Ines Meyer 22 Sook-Lei Liew 77 Florian Kofler 113388 Ivan Ezhov 113388 David
Robben 33991010 Alexander Hutton 77 Tassilo Friedrich 11 Teresa Zarth 11
Johannes Bürkle 11 The Anh Baran 11 Bjoern Menze 331111 Gabriel Broocks 44
Lukas Meyer 44 Claus Zimmer 11 Tobias Boeckh-Behrens 11 Maria Berndt 11 Benno
Ikenberg 1212 Benedikt Wiestler 1188 Jan S. Kirschke 1188

###### Abstract

Magnetic resonance imaging (MRI) is a central modality for stroke imaging. It
is used upon patient admission to make treatment decisions such as selecting
patients for intravenous thrombolysis or endovascular therapy. MRI is later
used in the duration of hospital stay to predict outcome by visualizing
infarct core size and location. Furthermore, it may be used to characterize
stroke etiology, e.g. differentiation between (cardio)-embolic and non-embolic
stroke. Computer based automated medical image processing is increasingly
finding its way into clinical routine. Previous iterations of the Ischemic
Stroke Lesion Segmentation (ISLES) challenge have aided in the generation of
identifying benchmark methods for acute and sub-acute ischemic stroke lesion
segmentation. Here we introduce an expert-annotated, multicenter MRI dataset
for segmentation of acute to subacute stroke lesions. This dataset comprises
400 multi-vendor MRI cases with high variability in stroke lesion size,
quantity and location. It is split into a training dataset of $n=250$ and a
test dataset of $n=150$. All training data will be made publicly available.
The test dataset will be used for model validation only and will not be
released to the public. This dataset serves as the foundation of the ISLES
2022 challenge with the goal of finding algorithmic methods to enable the
development and benchmarking of robust and accurate segmentation algorithms
for ischemic stroke.

###### Keywords:

Ischemic Stroke Lesion segmentation Multimodal MRI

##  1 Background & Summary

Stroke is a leading cause of morbidity and mortality worldwide [23]. Up to two
thirds of stroke survivors suffer permanent disability [13]. In the last
decade, the advent of endovascular reperfusion therapy has significantly
improved stroke outcome in patients with large vessel occlusions [5, 15, 19,
34]. Image-based guidance of revascularization treatment decisions has further
improved patient outcome for computer tomography (CT) [1, 6, 25] and magnetic
resonance imaging (MRI) [17, 41]. Computer aided image analysis, especially
for CT perfusion data has already found entry into clinical routine in many
centers and is recommended by national guidelines [32] to aid decision making
regarding reperfusion therapy [30, 44, 28, 31]. Deep learning approaches have
been shown to facilitate clinical interpretation of CT perfusion data [8, 16,
33, 10, 11]. Segmentation based volumetric analyses of stroke lesions in
magnetic resonance imaging (MRI) are often performed for research purposes and
have been shown to predict ischemia outcome [27, 47, 14]. However, stroke
lesion segmentations are usually painstakingly performed by hand and the
quality of annotations are heavily dependent on preexisting neuroimaging
experience of the rater and the total time and effort invested. The time-
consuming nature of this task prevents the regular use of segmentations during
clinical routine, which is further impeded by the high inter-observer
variability. Automated annotation of stroke lesions could be used in clinical
routine to guide therapeutic decisions in an acute setting and to predict
outcome at the subacute to chronic stage. Stroke lesion segmentation could
also be used to automatically classify stroke etiology in post-stroke MRI.

The first Ischemic Stroke Lesion Segmentation (ISLES) challenge, which took
place in 2015, was split into two sub-challenges: Sub-acute Stroke Lesion
Segmentation (SISS) and Stroke Perfusion Estimation. The goal of SISS (with a
total of 64 cases for training and testing) was to segment subacute stroke
lesions using conventional post-stroke MRI sequences, including T2 and T1
weighted imaging, fluid attenuated inversion recover (FLAIR), and diffusion
weighted imaging (DWI) [26]. The ISLES challenge 2018, its second iteration,
was set up to predict infarct core in DWI using CT perfusion data [16]. Both
ISLES events received major attention from the research community: there were
120 database downloads until the ISLES15 challenge day with 14 participating
teams, and the number of participating teams was roughly duplicated in the
latest ISLES’18 edition. The ISLES’15 and ISLES’18 challenges played a crucial
role in identifying prominent methods for acute and sub-acute ischemic stroke
lesion segmentation. These datasets have since served as important benchmarks
for the scientific community.

Based on the experience gained from these previous editions, ISLES’22 aims to
benchmark acute and sub-acute ischemic stroke MRI segmentation using 400
cases. This MICCAI 2022 challenge edition is organized with the task of DWI
infarct segmentation in acute and sub-acute stroke. ISLES’22 differs in
several ways from the previous challenge editions in ischemic stroke by: 1)
targeting the delineation of not only large infarct lesions, but also of
multiple embolic and/or cortical infarcts (typically seen after mechanical
recanalization), and 2) by evaluating both pre- and post- interventional MRI
images. Altogether, ISLES’22 provides a considerably larger dataset than
before (more than 5x the caseload than in the ISLES’15 edition). For detailed
information about the ISLES’22 challenge event, readers are referred to [9].
This challenge is held together with the ATLAS challenge (https://atlas.grand-
challenge.org/ [22] using the web-based platforms http://www.isles-
challenge.org/ and https://isles22.grand-challenge.org/.

From a clinical perspective, ISLES’22 focuses on the clinical growing interest
of acute embolic infarct patterns, both pre-intervention (i.e. at a very early
disease state) and post-intervention (typical, post-interventional sub-acute
infarct patterns not restricted to a single vessel territory). This clinical
problem also challenges the participants from a technical perspective: teams
will deal with a wider ischemic stroke disease spectrum, involving variable
lesion size and burden, more complex infarct patterns and variable
anatomically located lesions in data from multiple centers. The diversity of
the ISLES’22 dataset will provide a unique challenge for participants.

##  2 Methods

###  2.1 Ethical Statement

This retrospective evaluation of imaging data was approved by the local ethics
boards of all participating centers.

###  2.2 Subject selection

Inclusion criteria for the dataset: Subjects 18 years or older who had
received MR imaging of the brain for previously diagnosed or suspected stroke
were included in this study. The imaging protocol required at least a FLAIR
and DWI sequence. DWI consists of a trace image at a b-value up to 1000 s/mm²
as a well as its corresponding apparent diffusion coefficient (ADC) map. Image
acquisition was performed on one of the following devices: 3T Philips MRI
scanners (Achieva, Ingenia), 3T Siemens MRI scanner (Verio) or 1.5T Siemens
MAGNETOM MRI scanners (Avanto, Aera). MRIs included were intentionally chosen
to be heterogeneous to ensure the best possible training and generalization of
the algorithms. See Table 1 for an overview of MRI acquisition parameters.
Images were obtained by healthcare professionals as part of the clinical
imaging routine for stroke patients at three different stroke centers.

Table 1: Overview of MRI imaging parameters. Range of values [Min-Max] are shown. TR: Repetition time; TE: Echo time; TI: Inversion time. Modality | FLAIR | DWI
---|---|---
TR (ms) | [4800-12000] | [3175-16439]
TE (ms) | [103-395] | [55-91]
TI (ms) | [1650-2850] | -
Flip Angle ($\circ$) | [90-180] | 90
Voxel (mm2) | [0.23x0.23-1x1] | [0.88x0.88-2x2]
Slice thickness (mm) | [0.7-9.6] | [2.0-6.5]
Gap (mm) | [0.7-6.5] | [2.0-5.0]

Care was taken to select a broad spectrum of infarct patterns. All vascular
territories were included at a similar rate. A large subset of patients with
posterior circulation ischemia (e.g. due to basilar artery occlusion) was
included in this study. Due to a high degree of closely situated scull-base
artefacts in DWI, infratentorial ischemias are difficult to segment for
unexperienced raters and are frequently overlooked by segmentation algorithms
with little training exposure to posterior circulation ischemia. These
additional labeling challenges lead us to include more cases of posterior
circulation ischemia than would be expected if case selection were truly
random. Figure 1 shows sample cases portraying the spectrum of ischemia
included in this dataset.

Figure 1: Exemplary 3D Snapshots through the ischemic center of mass. Axial
FLAIR and DWI images are displayed in the top leftmost columns. The 3rd column
from the left shows the maximum intensity projection (MIP) of the mask (Msk).
The three rightmost columns show DWI with mask overlay (without MIP) in the
three anatomical planes. Top row: Larger left-sided infarct with bilateral
punctiform, likely embolic satellite ischemias. 2nd row: Infarct of the entire
right sided middle cerebral artery territory. 3rd row: Bilateral cerebellar
and occipital infarcts after posterior circulation ischemia. Bottom row:
Bilateral punctiform infarcts, likely resulting from multiple micro-embolic
occlusions.

Segmentation difficulties also frequently arise in cases with large amounts of
punctiform infarcts. For human raters, to adequately capture the entire
ischemic territory, a time-consuming segmentation effort is required.
Similarly, machine trained algorithms frequently fail to capture all the small
affected regions. In $n=5$ patients, where MR imaging was acquired for
suspected stroke, no ischemia was found. We chose to include this small subset
to diversify the dataset. Table 2 gives an overview of infarct volumes per
scan and per lesion, as well as the number of unconnected ischemias per scan
in the entire dataset. The number of unconnected ischemias has been calculated
using the python library cc3d [40].

Table 2: Overview of MRI imaging parameters. Range of values are shown. TR: Repetition time; TE: Echo time; TI: Inversion time. |  Scan infarct volume (ml) |  Number of unconnected ischemias |  Lesion-wise infarct volume (ml)
---|---|---|---
Mean (std) | 22.532 (44.756) | 9.298 (13.496) | 2.423 (15.804)
[Min, Max] | [0.000, 482.152 ] | [0, 126] | [0.003, 477.264 ]

In the hyper-acute phase of ischemic stroke, up to 4.5 hours post onset,
restricted diffusion is present (high signal on DWI and low signal on ADC)
while the FLAIR in the affected parenchyma remains without changes. This
imaging phenomenon is called a FLAIR-DWI mismatch and is used in clinical
practice to estimate the time window in patients where the time of onset is
unknown. An accurate estimate of the time of onset is crucial to make
decisions regarding revascularization treatment [41]. In acute ischemic
stroke, following the hyper-acute phase and usually defined in literature as 0
to 7 days from onset, DWI and FLAIR show a high signal with reduced ADC values
in the affected brain parenchyma. In the subacute stage, between 1 to 3 weeks
post onset, high DWI signal begins to diminish while ADC first normalizes to
values of healthy brain tissue, a phenomenon frequently referred to as
pseudonormalization. FLAIR signal remains high. In the chronic stage,
beginning 3 weeks after onset, DWI signal is variable but usually iso- to
hypo-intense depending on underlying T2 signal, while ADC values are high [2,
21, 42, 43, 24, 35, 29, 38, 45, 4]. MRIs with late acquisition post stroke
($>$ 1 week) often lead to a decreased DWI signal intensity for ischemic brain
parenchyma. A lower signal intensity in DWI leads to lower MRI sensitivity for
stroke and segmentation difficulty for both human and machine raters. In these
cases, it is especially difficult to adequately annotate the border between
ischemia and healthy brain tissue. This dataset includes cases of MRIs in
various stage of sub-acute stroke from multiple previous studies to find
machine learning solutions to this frequent issue in stroke lesion
segmentation [36, 37].

###  2.3 Ground truth stroke lesion segmentation

A hybrid human-algorithm annotation scheme was applied. First, the MR input
data was anonymized by conversion to Neuroimaging Informatics Technology
Initiative (NIfTI) format (https://nifti.nimh.nih.gov/nifti-1), according to
the Brain Imaging Data Structure (BIDS) convention
(https://bids.neuroimaging.io/). As part of pre-processing, DWI and its
corresponding ADC map were resliced using ANTs [3] to an axial isotropic voxel
size of 2$\times$2 $mm^{2}$. This was performed only for the cases of Center
#1 (see below), as this represents the original acquisition voxel size. The
slice thickness remained as acquired. FLAIR data remained as exported. A 3D
UNet [7] was trained on DWI data that was pre-annotated for other research
projects by using a single MRI modality (B=1000 DWI). This algorithm was
trained in-house with labeled data from University Hospital of Munich. Later,
this algorithm pre-segmented all scans intended for later release. All
algorithmic segmentations were then checked and edited by medical students
with special stroke lesion segmentation training. Their annotations were in
turn revised by a neuroradiology resident. Due to inadequate performance of
the primary pre-segmentation algorithm, largely due to the reasons mentioned
above, the original training data was critically revised and a large part was
deemed inadequate and discarded. A new pre-segmentation algorithm was trained
on annotations deemed correct or edited by the neuroradiologist in training.
This resulted in more accurate predictions and a lesser effort of correcting
annotations by medical students. All medical-student edited annotations were
critically revised by the neuroradiologist in training and a large part of the
final data set were reviewed by one out of three attending neuroradiologists,
all of them with more than 10 years of experience in stroke imaging. Manual
stroke lesion segmentations were performed using ITK-Snap or 3D Slicer open-
source tools for brain imaging visualization and segmentation [46, 12]. Figure
2 shows an overview of the annotation workflow followed in preparation of the
release of the dataset.

Figure 2: Workflow of the hybrid human-algorithm stroke lesion segmentation
applied in this dataset.

###  2.4 Post-annotation Data Pre-processing

In preparation for data-set release and in accordance to the ethical approval
obtained for this challenge, the imaging data was irreversibly anonymized. For
this de-identification of patients, brain-extraction was performed mask-based
using the HD-BET algorithm [18] completion of the annotation process. FLAIR to
DWI rigid registration was performed using Elastix [20, 39] and subsequent
skull-stripping of DWI and ADC using the registered brain mask was performed.
After skull stripping for de-identification, all imaging sequences were
returned to their native space before data release.

###  2.5 Data Records

Data structure and file formats. All medical imaging files were exported from
the Picture Archiving and Communication System in the NIfTI format.
Segmentation masks are also saved in NIfTI format. All data in the ISLES’22
dataset was separated into a training dataset (250 subjects) and a test
dataset (150 subjects). Corresponding scanner metadata from the Digital
Imaging and Communications in Medicine (DICOM) header in the JSON file format
is provided with the datasets if available. MRIs from the following centers
were included:

  * $\ast$

Center #1: University Hospital of the Technical University Munich, Munich,
Germany.

  * $\ast$

Center #2: University Hospital of Bern, Bern, Switzerland.

  * $\ast$

Center #3: University Medical Center Hamburg-Eppendorf, Hamburg, Germany.

The train set comprises data from centers #1 and #2. The test set comprises
data from all the three centers in equal parts as follows: Acute to early sub-
acute stroke data from centers #1 and #3 (MRIs acquired after
revascularization therapy). Hyper-acute to acute stroke data from center #2
(MRIs acquired before revascularization therapy). Thus, in this ISLES’22 task
we will evaluate the robustness and generalization capability of the proposed
models over 1) new scans coming from two centers already used at training
stage, 2) new scans coming from a new (unseen at training stage) center, and
3) new scans acquired before revascularization therapy, coming from a center
already seen at training stage. The split between training and test data set
will be done so that both sets will include a similar variance of stroke
lesion patterns ranging from large territorial infarcts to small punctiform
ischemia.

###  2.6 Technical Validation

The presented medical imaging data was derived from the picture archiving and
communication system of the corresponding institution and therefore fully
complies with the legal standards and quality controls for the acquisition of
medical imaging in Germany, European Union and Switzerland, as well as the
industrial standards of the scanner vendors. Segmentation masks were prepared
and annotated at voxel-level by a human-machine hybrid algorithm with
hierarchical manual checks and corrections first by specifically trained
medical students and later a neuroradiology resident. Afterwards, the masks
were reviewed, corrected, and finally approved by an expert neuroradiologist.

###  2.7 Data and Code Availability

The ISLES 2022 data can be downloaded by following the steps described at
https://isles22.grand-challenge.org/dataset/. This data is published under the
creative commons license CC BY-SA 4.0. Upon reasonable request, respecting
patient confidentiality and ethical standards, special arrangements for use
and re-distribution can be made directly with the authors. In order to
facilitate future users of this dataset to get familiarized with the images,
we have released the following ISLES 2022 Github repository:
https://github.com/ezequieldlrosa/isles22. The repository contains scripts to
read the images, visualize them, and to quantify the algorithmic results
performance with the same metrics used in the challenge to rank participants.

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