Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states
The hydrodynamic equations for indirect excitons in the double quantum wells are studied taking into account 1) a possibility of an exciton condensed phase formation, 2) the presence of pumping, 3) finite value of the exciton lifetime, 4) exciton scattering by defects. The threshold pumping emergenc...
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| Цитувати: | Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states / V.I. Sugakov // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33702:1-10. — Бібліогр.: 38 назв. — англ. |
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Sugakov, V.I. 2019-06-14T10:39:19Z 2019-06-14T10:39:19Z 2014 Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states / V.I. Sugakov // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33702:1-10. — Бібліогр.: 38 назв. — англ. 1607-324X PACS: 71.35.Lk, 73.21.Fg DOI:10.5488/CMP.17.33702 arXiv:1412.1799 https://nasplib.isofts.kiev.ua/handle/123456789/153500 The hydrodynamic equations for indirect excitons in the double quantum wells are studied taking into account 1) a possibility of an exciton condensed phase formation, 2) the presence of pumping, 3) finite value of the exciton lifetime, 4) exciton scattering by defects. The threshold pumping emergence of the periodical exciton density distribution is found. The role of localized and free exciton states is analyzed in the formation of emission spectra. Проведено аналiз рiвнянь гiдродинамiки екситонiв у квантовiй ямi. Рiвняння враховують 1) можливiсть фазового переходу в системi, 2) присутнiсть зовнiшньої накачки, 3) скiнчений час життя екситонiв, 4) розсiяння екситонiв на дефектах. Визначно порогову накачку утворення перiодичного розподiлу екситонної густини. Дослiджується вплив локалiзованих i вiльних екситонiв на формування спектрiв випромiнювання. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states Конденсацiя екситонiв в квантових ямах. Гiдродинамiка екситонiв. Вплив локалiзованих станiв дефектiв Article published earlier |
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Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states |
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Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states Sugakov, V.I. |
| title_short |
Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states |
| title_full |
Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states |
| title_fullStr |
Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states |
| title_full_unstemmed |
Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states |
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exciton condensation in quantum wells. exciton hydrodynamics. the effect of localized states |
| author |
Sugakov, V.I. |
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Sugakov, V.I. |
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2014 |
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English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
| title_alt |
Конденсацiя екситонiв в квантових ямах. Гiдродинамiка екситонiв. Вплив локалiзованих станiв дефектiв |
| description |
The hydrodynamic equations for indirect excitons in the double quantum wells are studied taking into account 1) a possibility of an exciton condensed phase formation, 2) the presence of pumping, 3) finite value of the exciton lifetime, 4) exciton scattering by defects. The threshold pumping emergence of the periodical exciton density distribution is found. The role of localized and free exciton states is analyzed in the formation of emission spectra.
Проведено аналiз рiвнянь гiдродинамiки екситонiв у квантовiй ямi. Рiвняння враховують 1) можливiсть
фазового переходу в системi, 2) присутнiсть зовнiшньої накачки, 3) скiнчений час життя екситонiв, 4) розсiяння екситонiв на дефектах. Визначно порогову накачку утворення перiодичного розподiлу екситонної
густини. Дослiджується вплив локалiзованих i вiльних екситонiв на формування спектрiв випромiнювання.
|
| issn |
1607-324X |
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https://nasplib.isofts.kiev.ua/handle/123456789/153500 |
| citation_txt |
Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states / V.I. Sugakov // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33702:1-10. — Бібліогр.: 38 назв. — англ. |
| work_keys_str_mv |
AT sugakovvi excitoncondensationinquantumwellsexcitonhydrodynamicstheeffectoflocalizedstates AT sugakovvi kondensaciâeksitonivvkvantovihâmahgidrodinamikaeksitonivvplivlokalizovanihstanivdefektiv |
| first_indexed |
2025-11-25T23:08:43Z |
| last_indexed |
2025-11-25T23:08:43Z |
| _version_ |
1850578874092486656 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 3, 33702: 1–10
DOI: 10.5488/CMP.17.33702
http://www.icmp.lviv.ua/journal
Exciton condensation in quantum wells. Exciton
hydrodynamics. The effect of localized states
V.I. Sugakov
Institute for Nuclear Research, 47 Nauky ave., 03680 Kyiv, Ukraine
Received April 29, 2014
The hydrodynamic equations for indirect excitons in the double quantum wells are studied taking into account 1)
a possibility of an exciton condensed phase formation, 2) the presence of pumping, 3) finite value of the exciton
lifetime, 4) exciton scattering by defects. The threshold pumping emergence of the periodical exciton density
distribution is found. The role of localized and free exciton states is analyzed in the formation of emission
spectra.
Key words: self-organization, quantum wells, excitons, phase transition
PACS: 71.35.Lk, 73.21.Fg
1. Introduction
Phase transitions in systems of unstable particles are specific examples of non-equilibrium phase
transitions and processes of self-organization [1]. In such a system, particles are created by external
sources and disappear due to different reasons. If there is an attractive interaction between the parti-
cles, they may create a condensed phase during their lifetime. A steady state may arise in a system if
the number of the created particles in the unit time is equal to the number of the disappeared particles.
This state is stationary, but it is not equilibrium. The following examples of such systems with unstable
particles may be presented: 1) dielectric exciton liquid in crystals; 2) electron-hole liquid in semiconduc-
tors; 3) highly excited gas with many excited molecules; 4) vacancies and interstitials in a cryatal created
by nuclear irradiation; 5) quark glyuon plasma and others. The finite value of the particles determine
some peculiarities of phase transitions in such systems. The main peculiarities are as follows: a) a phase
transition in a system of unstable particles may occur if the lifetime is larger than some critical value;
b) in the presence of parameters at which two phases coexistent, the sizes of the regions of condensed
phases of unstable particles are restricted: c) there is strong spatial correlation between different regions
of condensed phases, that is why periodical structures may arise.
The present paper is devoted to an investigation of self-organization processes of the exciton system
in semiconductor quantum wells. The appearance of periodical dissipative structures in exciton systems
at the irradiations greater than some critical value was shown in the work [2, 3]. Experimental obser-
vation of a periodical distribution of the exciton density was obtained in [4, 5] in a system of indirect
excitons in semiconductor double quantum wells. An indirect exciton consists of an electron and hole
separated over two wells by an electric field. Due to the damping of the electron-hole recombination, in-
direct excitons have a large lifetime which makes it possible to create a high concentration of excitons at
small pumping and to study the manifestation of the effects of exciton-exciton interaction. The authors
of the paper [4] observed the emission from a double quantum well on the basis of AlGaAs system in
the form of periodically situated islands along the ring around the laser spot. In the paper [5], in which
the excitation of a quantum well was carried out through a window in a metallic electrode, a periodi-
cal structure of the islands situated along the ring under the perimeter of the window was found in the
luminescence spectrum. The islands were observed at a frequency that corresponds to the narrow line
arising at a threshold value of pumping [6]. Afterwards, spatial structures of exciton density distributions
© V.I. Sugakov, 2014 33702-1
http://dx.doi.org/10.5488/CMP.17.33702
http://www.icmp.lviv.ua/journal
V.I. Sugakov
were observed in a single wide quantum well [7], in different types of electrodes that create a periodical
potential [8] or have windows in the shape of a rectangle, two circles and others [9, 10].
The phenomenon of a symmetry loss and the creation of structures in the emission spectra of indirect
excitons urged a series of theoretical investigations [11–18]. The main efforts were directed towards the
verification of a fundamental possibility of the appearance of the periodicity of the exciton density distri-
bution. A specific explanation of the experiment is presented in two works [11, 16] with respect to only
one experiment [4].The authors of the work [11] considered the instability that arises under the kinetics
of level occupations by the particles with the Bose-Einstein statistics. Namely, the growth of the occupa-
tion of the level with zero moment should stimulate the transitions of excitons to this level. However, the
density of excitons was found greater, and the temperature was found lower than these values observed
in the experiments. In the paper [16] the authors did not take into account the screening between the
charges at macroscopic distances.
Our model is based on the appearance of self-organization processes in non-equilibrium systems for
excitons with attractive interaction shown in [2, 3]. Investigations performed in this model [19–25] gave
the explanations of spatial structures and their temperature and pumping dependencies obtained in dif-
ferent experiments [4, 5, 8–10]. Theoretical approaches of the works [19–25] are based on the following
assumptions.
1. There is an exciton condensed phase caused by the attractive interaction between excitons. The
existence of attractive interaction between excitons is confirmed by the calculations of biexcitons
[26–29], and by investigations of amany-exciton system [30]. Nevertheless, there is an experimental
work [31], where the authors explain their experimental results by the existence of a repulsion
interaction between excitons. These results come into conflict with our suggestion regarding the
attractive interaction. We shall remove this contradiction in section 3.
2. The finite value of the exciton lifetime plays an important role in the formation of a spatial distribu-
tion of exciton condensed phases. As usual, the exciton lifetime significantly exceeds the duration
of the establishment of a local equilibrium. However, it is necessary to take into account the finite-
ness of the exciton lifetime in the study of spatial distribution phases in two-phase systems, because
the exciton lifetime is less than the time of the establishment of equilibrium between phases. The
latter is determined by slow diffusion processes and is long.
Two approaches of the theory of phase transitions were used while developing the theory: the model
of nucleation (Lifshits-Slyozov) and the model of spinodal decomposition (Cahn-Hillart). These models
were generalized to the particles with the finite lifetime, which is important for interpretation of the
experimental results. The involvement of Bose-Einstein condensation for excitons was not required in
order to explain the experiments.
In the present paper, the hydrodynamic equation for excitons is investigated for the case of excitons
being in a condensed phase. The appearance of an instability of the uniform distribution of the exciton
density and the development of nonhomogeneous structures are studied. The effect of defects on spectral
positions of the emission spectra of both gas and condensed phases is analysed as well.
2. Analysis of hydrodynamic equations of exciton condensed phase
Hydrodynamic equations of excitons were obtained and analysed in the work [32]. Hydrodynamic
equations of excitons generalizing the Navier-Stokes equations that take into account the finite exciton
lifetime, the pumping of exciton, the existence of an exciton condensed phase and the presence of defects
were developed in [33]. In the paper, we make some analysis of these equations.
The system is described by the exciton density n ≡ n(~r , t) and by the velocity of the exciton liquid
~u ≡~u(~r , t). The equations for conservation of the exciton density and for themovement of exciton density
33702-2
Exciton condensation
are basic for the exciton hydrodynamic equations.
∂n
∂t
+div(n~u) =G −
n
τex
, (2.1)
∂mnui
∂t
=−
∂Πik
∂xk
−
mnui
τsc
, (2.2)
where G is the pumping (the number of excitons created for unit time in unit area of the quantum well),
τex is the exciton lifetime,m is the exciton mass, Πik is the tensor of density of the exciton flux
Πik = Pik +mnui uk −σ′
ik , (2.3)
where Pik is the pressure tensor, σ′
ik
is the viscosity tensor of tension. In the equation (2.2), we neglected
the small momentum change caused by the creation and the annihilation of excitons.
Introducing coefficients of viscosity and using (2.1), equation (2.2) may be rewritten in the form
ρ
[
∂ui
∂t
+
(
uk
∂
∂xk
)
ui
]
=−
∂Pik
∂xk
+η∆ui + (ς+η/3)
(
∂
∂xi
)
div~u−
ρui
τsc
, (2.4)
where ρ = mn is the mass of excitons in the unit volume.
We assume that the state of the local equilibrium is realized and the state of the system may be de-
scribed by free energy, which depends on a spatial coordinate. Let us present the functional of the free
energy in the form
F =
∫
d~r
[
K
2
(
~∇n
)2
+ f (n)
]
. (2.5)
At the given presentation of free energy, the pressure tensor is determined by the formula [34]
Pαβ =
[
p −
K
2
(
~∇n
)2
−K n∆n
]
δαβ+K
∂n
∂xα
∂n
∂xβ
, (2.6)
where p = n f ′(n)− f (n) is the equation of the state, p is the isotropic pressure.
Taking into account (2.6) and introducing coefficients of viscosity, we finally rewrite the equation (2.2)
in the form
∂ui
∂t
+uk
∂ui
∂xk
+
1
m
∂
∂xi
(
−K∆n+
∂ f
∂n
)
+ν∆ui + (ς/m +ν/3)
(
∂
∂xi
)
div~u+
ui
τsc
= 0. (2.7)
Equations (2.1), (2.7) are the equations of the hydrodynamics for an exciton system. It follows from the
estimations, made in the work [32], that the terms with the viscosity coefficients are small and we shall
neglect them.
In the case of a steady state irradiation, the equations (2.1) and (2.7) have the solution n = Gτ, u =
0. To study the stability of the uniform solution we consider, that the behavior of a small fluctuation
of the exciton density and the velocity from these values are as follows: n → n +δn exp[i~k ·~r +λ(~k)t ],
u = δu exp[i~k ·~r +λ(~k)t ]. Having substituted these expressions in equations (2.1), (2.7), we obtain, in the
linear approximation with respect to fluctuations, the following expression
λ±(~k) =
1
2
−
(
1
τsc
+
1
τex
)
±
√
(
1
τsc
−
1
τex
)2
−
4k2n
m
(
k2K +
∂2 f
∂n2
)
. (2.8)
It follows from (2.8) that both parameters λ±(~k) have a negative real part at small and large values
of vector ~k and, therefore, the uniform solution of the hydrodynamic equation is stable. The instability
with respect to a formation of nonhomogeneous structures arises at some threshold value of exciton
density and at some critical value of the wave vector, when ∂2 f /∂n2 becomes negative. The analysis of
the equation (2.8) gives the following expression for the critical values of the wave vector kc and the
exciton density nc
k4
c =
m
K ncτscτex
, (2.9)
k2
c nc
m
(
k2
c K +
∂2 f (nc)
∂n2
c
)
+
1
τscτex
= 0. (2.10)
33702-3
V.I. Sugakov
For stable particles (τex →∞), the equations (2.9), (2.10) give the condition ∂2 f /∂n2
= 0, which is the
condition for spinodal decomposition for a system in the equilibrium case.
Depending on parameters, the equations (2.1), (2.7) describe the ballistic and diffusion movement of
the exciton system. The relaxation time τsc plays an important role in the formation of the exciton move-
ment. Due to the appearance of nonhomogeneous structures, there exist exciton currents in a system
(~j = n~u , 0) even under the uniform steady-state pumping. Excitons are moving from the regions hav-
ing a small exciton density to the regions having a high density. In the present 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 assume the existence of the following
conditions
∂ui
∂t
≪ ui /τsc , (2.11)
uk
∂ui
∂xk
≪ ui /τsc . (2.12)
Particularly, the equation (2.11) holds in the study of the steady-state exciton distribution. The fulfilment
of equation (2.12) will be shown later following some numerical calculations.
Using the conditions (2.11) and (2.12), we obtain from equation (2.7) the value of the velocity ~u
~u =−
τsc
m
~∇
(
−K∆n+
∂ f
∂n
)
. (2.13)
As a result, the equation for the exciton density current may be presented in the form
~j = n~u =−M∇µ, (2.14)
where µ= δF /δn is the chemical potential of the system, M = nD/κT is the mobility, D =κTτsc/m is the
diffusion coefficient of excitons.
Therefore, the equation for the exciton density (2.1) equals
∂n
∂t
=
D
κT
(
−K n∆2n−K~∇n ·~∇∆n
)
+
D
κT
~∇·
(
n
∂2 f
∂n2
~∇n
)
+G −
n
τex
. (2.15)
Just in the form of (2.15), we investigated a spatial distribution of the exciton density at exciton con-
densation using the spinodal decomposition approximation by choosing different dependencies f on n
[21, 23, 25]. Thus, our previous consideration of the problem corresponds to the diffusion movement
of hydrodynamic equations (2.1), (2.7). For the system under study, a condensed phase appears if the
function f (n) describes a phase transition. In the papers mentioned above, the examples of such depen-
dencies were given. Here, we analyse another dependence f (n), which is also often used in the theory of
phase transitions
f =κT n [ln(n/na )−1]+a
n2
2
+b
n4
4
+c
n6
6
, (2.16)
where a, b, c are constant values. Three last terms in the formula (2.16) 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 (this is important if
such a region exists in a system). At an increase of the exciton density, the term an2/2 manifests itself
firstly. It contributes the an value to the chemical potential. In our system, the origin of this term is
connected with the dipole-dipole interaction, which should become apparent at the beginning with the
growth of the density due to its long-range nature. To estimate a for the dipole-dipole exciton interaction
in double quantum well we may use the plate capacitor formula an = 4πe2dn/ǫ, where d is the distance
between the wells, ǫ is the dielectric constant. This formula is usually used to determine the exciton
density from the experimental meaning of the blue shift of the frequency of the exciton emission with
the rise of the density. It follows from the formula that a = 4πe2d/ǫ. When the exciton density grows,
the last two terms in (2.14) begin to play a role. The existence of a condensed phase requires that the
value b should be negative (b < 0). For stability of a system, at large n, the parameter c should be positive
33702-4
Exciton condensation
Figure 1. The spatial dependence of the exciton density at a different value of the pumping: for a contin-
ues line G = 0.0055, for a periodical line G = 0.008, for a dashed line G = 0.0092. D1 = 0.03, b =−1.9.
(c > 0). It is assumed in themodel that the condensed phase arises due to the exchange and Van derWaals
interactions. The calculations show that in some region of distances between the wells, these interactions
exceed the dipole-dipole repulsion.
Let us introduce dimensionless parameters: ñ = n/n0, where n0 = (a/c)1/4, b̃ = b/(ac)1/2, ~̃r =~r /ξ,
where ξ = (K /a)1/2 is the coherence length, t̃ = t/t0, where t0 = κT K /(Dn0a2), D1 = κT /(an0), G̃ =
Gt0/n0, τ̃ex = τ/t0. As a result, the equation (2.15) is reduced to the form (hereinafter the symbol ∼ will
be omitted in the equation)
∂n
∂t
= D1∆n−n∆2n+n∆n
(
1+3bn2
+5n4
)
−~∇n ·~∇∆n+
(
~∇n
)2 (
1+9bn2
+25n4
)
+G −
n
τex
. (2.17)
The solutions of the equation (2.17) are presented in figure 1 for the one-dimensional case [n(~r , t) ≡
n(z, t)] for three values of the steady-state uniform pumping.
The solutions are obtained at the initial conditions n(z,0) = 0 and the boundary conditions n′(0, t) =
n′(L, t) = n′′(0, t) = n′′(L, t) = 0, where L is the size of a system. The periodical solution exists in some
interval of the pumping Gc1 <G <Gc2. At specified parameters, the periodical solution exists at 0.0055 <
G < 0.0092. Outside this region, the solution describes a uniform system: the gas phase at a low pumping
and the condensed phase at a large pumping. The upper part of the periodical distribution corresponds
to a condensed phase, the lower part corresponds to the gas phase. The size of the condensed phase in-
creases with the change of the pumping from Gc1 to Gc2. Figure 2 shows the spatial dependence of the
exciton current calculated by the formula (2.14). 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. The results for the currents in figure 2 are presented in dimensionless
units: j̃ = j / j0, where j0 = n0u0, u0 = (τscn0a)/(mξ) is the unit of the velocity. The exciton density is
presented in figure 1 in dimensionless units (ñ = n/n0). It is seen in figure 1 that ñ ∼ 1, and the magni-
tude of n is of an order of n0. Thus, for estimations we may assume that n0a corresponds to the shift of
the luminescence line with an increase of the exciton density, the magnitude of ξ is of the order of the
size of the condensed phase. For the following magnitudes of parameters τsc = 10−11 s, n0a = 2 ·10−3 eV,
m = 2 · 10−28 g, ξ = 2 · 10−4 cm, we obtain u0 ∼ 106 cm/s. According to calculations (see figure 2), the
magnitudes of the current and the velocity are two orders of magnitude less than their units j0 and u0,
so the condition u ∼ 104 cm/s takes place. In order to verify the fulfilment of the condition (2.12), let us
suppose that (∂ui )/(∂xk ∼ u/l ), where l is the period of a structure. It follows from experiments [4, 5] that
l ∼ (5÷10) µm. Using these data we see that the condition (2.12) is very well satisfied. This condition is
violated at τsc Ê 10−9 s. Therefore, the formation of nonuniform exciton dissipative structures in a dou-
ble quantum well occurs due to the diffusion movement of excitons. To prove the main hydrodynamic
equation (2.15), the last term in equation (2.2)is of importance. It describes the loss of the momentum
33702-5
V.I. Sugakov
Figure 2. The spatial dependence of the exciton current at G = 0.008, D1 = 0.03, b =−1.9.
due to the scattering of excitons by defects and phonons. It is this term that describes the processes that
cause a decay of the exciton flux. From the viewpoint of a 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 latter systems, the phonons (movement of particles) are an intrin-
sic compound part of the system spectrum, the interaction between phonons (particles) is the interaction
between the atoms of a system and does not cause the change of the complete momentum of a system
and its movement as a whole. Phonons and defects for excitons are external subsystems that brake the
exciton movement. Therefore, to create the exciton superfluidity, it is needed that the value of τsc should
grow significantly. This is possible for exciton polaritons that weakly interact with phonons; moreover,
there is a certain experimental evidence on an observation of the polariton condensation [35]. For indi-
rect excitons, the critical temperature of a superfluid transition is strongly lowered by inhomogeneities
[36, 37]. Thus, the question regarding the possibility of the superfluidity existence for indirect excitons
on the basis of AlGaAs system is open.
Thus, the peculiarities observed at large densities of indirect excitons may be explained by phase
transitions in a system of particles having attractive interactions and by the finite value of the lifetime
without an involvement of the Bose-Einstein condensation.
3. Distribution of excitons between localized and delocalized states
According to the experimental results [31], the frequency of the emission from the islands of a con-
densed phase, where the exciton density is large, is higher than the frequency of emission from the region
between islands, in which the density is less. The authors made the conclusion [31] that the interaction be-
tween excitons is repulsive, and, therefore, the formation of a condensed phase by attractive interaction
is impossible. This contradicts the main assumption of our works [19–25], though these works explain
many experiments. Now, we remove this contradiction taking into account the presence of localized ex-
citons.
The localized states arise due to the presence of residual donors, acceptors, defects, and inhomoge-
neous thickness of the wells. Their existence is confirmed by the presence of an emission at the frequen-
cies less than the frequency of the exciton band emission and by broadening of exciton lines. At a low
temperature and at a small pumping, the main part of the emission band consists of the emission from
defect centers, while the part of the exciton emission grows with an increased pumping. Now, we con-
sider the relation between the contribution to the emission band intensity from free excitons and from
the excitons (pairs of electrons and holes) localized on the defects. We assume that the localized states
are saturable, namely, every center may capture a restricted number of electron-hole pairs. In our calcu-
lations we assume that only a single excitation may be localized on a defect. There are no other localized
excitations or they have a very low binding energy and are unstable. The dependence of the density of lo-
calized states on the energy was chosen in the exponential form, namely ρ(E ) =αNl exp(αE ), where Nl is
the density of the defect centers, E is the depth of the trap level. The exciton states (free and localized) are
33702-6
Exciton condensation
distributed onto levels after the creation of electrons and holes due to an external irradiation and their
subsequent recombination and relaxation. Since the time of relaxation is much less than the exciton life-
time, the distribution of excitation between free and localized states corresponds to the thermodynamical
equilibrium state. In the considered model, we should obtain a distribution of electron-hole pairs, whose
population on a single level may be changed from zero to infinity for E > 0 (for free exciton states) and
from zero to one for E < 0 (for localized states). Formally, in the considered system, free excitons have
Bose-Einstein statistics while localized excitations obey the Fermi-Dirac statistics. At a small exciton den-
sity, Bose-Einstein and Boltsmann statistics give similar results for free excitons, but the application of
Fermi-Dirac statistics for localized states on a single level for one trap is important. The equation for
energy distribution may be found from the minima of a large canonical distribution
w(nk ,ni ) = exp
(
Ω+Nµ−E
κT
)
, (3.1)
where N = Σi ni +Σk nk , E = Σi ni Ei +Σk ,l nk Ek , ni = 0,1, nk = 0,1, . . . ,∞, k is the wave vector of the
exciton, l designates the singular levels. Parameter µ is the exciton chemical potential.
The distribution of excitons over free and localized levels is determined from the minimum of the
functional (3.1). As a result, we obtain the following conditions for the mean values of the free exciton
density n and the density of localized states nL
nex =
gν
4πEexa2
ex
∞
∫
0
dE
exp
(
E−µ
κT
)
−1
, (3.2)
nL =αNl
0
∫
−∞
exp(αE )dE
exp
(
E−µ
κT
)
+1
, (3.3)
where aex = (ħ2ε)/(µexe2) and Eex = (µexe4)/(2ε2
ħ
2) are the radius and the energy of the exciton in the
ground state in the bulkmaterial, g = 4, µex is the reducedmass of the exciton, ν is the ratio of the reduced
and the total mass of the exciton. The chemical potential µ is determined from the condition
nL +n =Gτex , (3.4)
Figure 3. The dependence of the density of free (thick line) and trapped (thin line) excitons on the pump-
ing. The parameters of the system: T = 2K , Nl = 0.001/a2
ex ,α= 300(eV)−1.
33702-7
V.I. Sugakov
Figure 4. (Color online) The distribution of excitations in the traps and in the states of the exciton band.
The thick line in figure 4 (a) corresponds to the energy per a single exciton in the condensed phase. On
the right [figure 4 (b)], the upper line describes the whole emission from the island (the emission of both
the condensed phase and trapped excitons), the low line describes the emission of the trapped excitons.
where Gτex is the whole number of excitation (free and localized) per unit surface.
The dependence of distribution of free and localized excitons on the pumping is presented in figure 3
as a function of the whole number of excitation presented in units of 1/a2
ex. Let the exciton radius be
equal to 10 nm. Then, the concentration of the traps and the width of the distribution of trap levels,
chosen under calculations of figure 3, are of the order of 109 cm−2 and 0.003 eV, correspondingly.
As it is seen in figure 3, the number of localized excitations at small pumping exceeds the number of
free excitons and the emission band is determined by the emission from the traps. With an increase of
pumping, the occupation of the trap levels becomes saturated. For the chosen parameters, the concentra-
tion of excitations under saturation is of the order of 109 cm−2. The exciton density grows simultaneously
with the saturation of the localized levels. As a result, the shortwave part of the emission band should
increase with an increased pumping. When the exciton density becomes larger, the collective exciton
effects begin to manifest themselves. The equations (3.2), (3.3) do not take into account the interactions
between excitations, and special models and theories are needed to describe the collective effects. The
appearance of a narrow line was observed in [6] with an increased pumping on the shortwave part of the
exciton emission band. Simultaneously, patterns arise in the emission spectra. The narrow line appeared
after the localized states become occupied. According to [6], this line is explained by the exciton Bose-
Einstain condensation. According to our model [19, 22], the appearance of the islands corresponds to the
condensed phase caused by the attractive interaction between excitons. The energy per a single exciton
in the condensed phase is less than the energy of free excitons (the thick line in figure 4), but the gain of
the energy under condensation of indirect excitons in AlGaAs system is less than the whole bandwidth,
which are formed by the localized and delocalized states. Thus, the energy of photons emitted from the
islands of a condensed phase is higher than the energy of photons emitted by traps (see figure 4). The
excitons cannot leave the condensed phase (the islands) and move to the traps (to the states of lower
energy) since the levels of the traps are already occupied. This may be the reason of the results obtained
in [31], where the maximum of the frequency of emission from the islands is higher than the maximum
frequency from the regions between the islands in spite of the attractive interaction between the excitons.
The qualitative results coincide with the results obtained in [33] using another method from the so-
lution of kinetic equations for level distributions at some simple approximation for the probability tran-
sition between the levels. Similar behavior of distribution of free and trapped excitons is observed for
another energy dependence of the density of localized states.
The results may be used to explain the intensity and temperature dependencies of the exciton emis-
sion of dipolar excitons in InGaAs coupled double quantum wells [38]. The authors observed a growth of
the shortwave side of the band with an increased pumping.
4. Conclusion
Hydrodynamic equations are analyzed for excitons in a double quantum well. The equations take
into account the presence of pumping, the finite value of the exciton lifetime and the possibility of a
condensed phase formation in a phenomenological model. The equations describe the diffusion and the
ballistic movement of an exciton system. It is shown that the spatial nonuniform structures, observed
33702-8
Exciton condensation
experimentally in double wells on the basis of AlGaAs crystal, may be explained by hydrodynamic equa-
tions in the diffusion approximation. The effect of saturable localized states on spectral distribution of the
emission from condensed and gas phases is obtained. The theory explains the features of experimental
dependencies of the emission spectra from the condensed and gas phases.
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http://dx.doi.org/10.1134/S0021364012140056
http://dx.doi.org/10.1103/PhysRevLett.94.176404
http://dx.doi.org/10.1088/0953-8984/18/42/012
http://dx.doi.org/10.1016/j.physleta.2007.04.001
http://dx.doi.org/10.1134/S0021364007220158
http://dx.doi.org/10.1134/S1063783410020046
http://dx.doi.org/10.1103/PhysRevLett.109.187402
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http://dx.doi.org/10.1134/1.567264
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V.I. Sugakov
Конденсацiя екситонiв в квантових ямах. Гiдродинамiка
екситонiв. Вплив локалiзованих станiв дефектiв
В.Й. Сугаков
Iнститут ядерних дослiджень, просп. Науки, 47, 03680 Київ, Україна
Проведено аналiз рiвнянь гiдродинамiки екситонiв у квантовiй ямi. Рiвняння враховують 1) можливiсть
фазового переходу в системi, 2) присутнiсть зовнiшньої накачки, 3) скiнчений час життя екситонiв, 4) роз-
сiяння екситонiв на дефектах. Визначно порогову накачку утворення перiодичного розподiлу екситонної
густини. Дослiджується вплив локалiзованих i вiльних екситонiв на формування спектрiв випромiнюван-
ня.
Ключовi слова: самоорганiзацiя, квантовi ями, екситони, фазовий перехiд
33702-10
Introduction
Analysis of hydrodynamic equations of exciton condensed phase
Distribution of excitons between localized and delocalized states
Conclusion
|