Exciton condensation in quantum wells. Exciton hydrodynamics. The effect of localized states
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Exciton condensation in quantum wells. Excitonhydrodynamics. 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 excitonlifetime, 4) exciton scattering by defects. The threshold pumping emergence of the periodical exciton densitydistribution is found. The role of localized and free exciton states is analyzed in the formation of emissionspectra.
Key words: self-organization,quantumwells,excitons,phasetransition
PACS:
1. Introduction
Phase transitions in systems of unstable particles are specific examples of non-equilibrium phasetransitions and processes of self-organization [1]. In such a system, particles are created by externalsources 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 ifthe 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 unstableparticles 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 createdby nuclear irradiation; 5) quark glyuon plasma and others. The finite value of the particles determinesome peculiarities of phase transitions in such systems. The main peculiarities are as follows: a) a phasetransition 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 condensedphases of unstable particles are restricted: c) there is strong spatial correlation between different regionsof condensed phases, that is why periodical structures may arise.The present paper is devoted to an investigation of self-organization processes of the exciton systemin semiconductor quantum wells. The appearance of periodical dissipative structures in exciton systemsat 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 indirectexcitons in semiconductor double quantum wells. An indirect exciton consists of an electron and holeseparated 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 atsmall pumping and to study the manifestation of the effects of exciton-exciton interaction. The authorsof the paper [4] observed the emission from a double quantum well on the basis of AlGaAs system inthe form of periodically situated islands along the ring around the laser spot. In the paper [5], in whichthe 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 theluminescence spectrum. The islands were observed at a frequency that corresponds to the narrow linearising at a threshold value of pumping [6]. Afterwards, spatial structures of exciton density distributions © V.I. Sugakov, 2014 .I. Sugakov were observed in a single wide quantum well [7], in different types of electrodes that create a periodicalpotential [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 indirectexcitons urged a series of theoretical investigations [11–18]. The main efforts were directed towards theverification 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 onlyone experiment [4].The authors of the work [11] considered the instability that arises under the kineticsof 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, thedensity of excitons was found greater, and the temperature was found lower than these values observedin the experiments. In the paper [16] the authors did not take into account the screening between thecharges at macroscopic distances.Our model is based on the appearance of self-organization processes in non-equilibrium systems forexcitons with attractive interaction shown in [2, 3]. Investigations performed in this model [19–25] gavethe 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 followingassumptions.1. There is an exciton condensed phase caused by the attractive interaction between excitons. Theexistence of attractive interaction between excitons is confirmed by the calculations of biexcitons[26–29], and by investigations of a many-exciton system [30]. Nevertheless, there is an experimentalwork [31], where the authors explain their experimental results by the existence of a repulsioninteraction between excitons. These results come into conflict with our suggestion regarding theattractive 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 durationof 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, becausethe exciton lifetime is less than the time of the establishment of equilibrium between phases. Thelatter is determined by slow diffusion processes and is long.Two approaches of the theory of phase transitions were used while developing the theory: the modelof nucleation (Lifshits-Slyozov) and the model of spinodal decomposition (Cahn-Hillart). These modelswere generalized to the particles with the finite lifetime, which is important for interpretation of theexperimental results. The involvement of Bose-Einstein condensation for excitons was not required inorder to explain the experiments.In the present paper, the hydrodynamic equation for excitons is investigated for the case of excitonsbeing in a condensed phase. The appearance of an instability of the uniform distribution of the excitondensity and the development of nonhomogeneous structures are studied. The effect of defects on spectralpositions 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]. Hydrodynamicequations of excitons generalizing the Navier-Stokes equations that take into account the finite excitonlifetime, the pumping of exciton, the existence of an exciton condensed phase and the presence of defectswere 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 the movement of exciton density xciton condensation are basic for the exciton hydrodynamic equations. ∂ n ∂ t + div ( n ~ u ) = G − n τ ex , (2.1) ∂ mnu i ∂ t = − ∂ Π ik ∂ x k − mnu i τ sc , (2.2)where G is the pumping (the number of excitons created for unit time in unit area of the quantum well), τ ex is the exciton lifetime, m is the exciton mass, Π ik is the tensor of density of the exciton flux Π ik = P ik + mnu i u k − σ ′ ik , (2.3)where P ik is the pressure tensor, σ ′ ik is the viscosity tensor of tension. In the equation (2.2), we neglectedthe small momentum change caused by the creation and the annihilation of excitons.Introducing coe ffi cients of viscosity and using (2.1), equation (2.2) may be rewritten in the form ρ · ∂ u i ∂ t + µ u k ∂∂ x k ¶ u i ¸ = − ∂ P ik ∂ x k + η ∆ u i + ( ς + η /3) µ ∂∂ x i ¶ div ~ u − ρ u i τ sc , (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 freeenergy in the form F = Z d ~ r · K ¡ ~ ∇ n ¢ + f ( n ) ¸ . (2.5)At the given presentation of free energy, the pressure tensor is determined by the formula [34] P αβ = · p − K ¡ ~ ∇ n ¢ − K n ∆ n ¸ δ αβ + K ∂ n ∂ x α ∂ n ∂ x β , (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 coe ffi cients of viscosity, we fi nally rewrite the equation (2.2)in the form ∂ u i ∂ t + u k ∂ u i ∂ x k + m ∂∂ x i µ − K ∆ n + ∂ f ∂ n ¶ + ν ∆ u i + ( ς / m + ν /3) µ ∂∂ x i ¶ div ~ u + u i τ sc = (2.7)Equations (2.1), (2.7) are the equations of the hydrodynamics for an exciton system. It follows from theestimations, made in the work [32], that the terms with the viscosity coe ffi cients are small and we shallneglect them.In the case of a steady state irradiation, the equations (2.1) and (2.7) have the solution n = G τ , u = . To study the stability of the uniform solution we consider, that the behavior of a small fl uctuationof the exciton density and the velocity from these values are as follows: n → n + δ n exp[i ~ k · ~ r + λ ( ~ k ) t ] , u = δ u exp[i ~ k · ~ r + λ ( ~ k ) t ] . Having substituted these expressions in equations (2.1), (2.7), we obtain, in thelinear approximation with respect to fl uctuations, the following expression λ ± ( ~ k ) = − µ τ sc + τ ex ¶ ± sµ τ sc − τ ex ¶ − k nm µ k K + ∂ f ∂ n ¶ . (2.8)It follows from (2.8) that both parameters λ ± ( ~ k ) have a negative real part at small and large valuesof vector ~ k and, therefore, the uniform solution of the hydrodynamic equation is stable. The instabilitywith respect to a formation of nonhomogeneous structures arises at some threshold value of excitondensity and at some critical value of the wave vector, when ∂ f / ∂ n becomes negative. The analysis ofthe equation (2.8) gives the following expression for the critical values of the wave vector k c and theexciton density n c k c = mK n c τ sc τ ex , (2.9) k c n c m µ k c K + ∂ f ( n c ) ∂ n c ¶ + τ sc τ ex = (2.10) .I. Sugakov For stable particles ( τ ex → ∞ ), the equations (2.9), (2.10) give the condition ∂ f / ∂ n = , which is thecondition 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 ofthe exciton system. The relaxation time τ sc 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 , ) 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 considerthe 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 followingconditions ∂ u i ∂ t ≪ u i / τ sc , (2.11) u k ∂ u i ∂ x k ≪ u i / τ sc . (2.12)Particularly, the equation (2.11) holds in the study of the steady-state exciton distribution. The ful fi lmentof 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 = − τ sc m ~ ∇ µ − 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 = δ F / δ n is the chemical potential of the system, M = nD / κ T is the mobility, D = κ T τ sc / m is thediffusion coe ffi cient of excitons.Therefore, the equation for the exciton density (2.1) equals ∂ n ∂ t = D κ T ¡ − K n ∆ n − K ~ ∇ n · ~ ∇ ∆ n ¢ + D κ T ~ ∇ · µ n ∂ f ∂ n ~ ∇ n ¶ + G − n τ ex . (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 movementof hydrodynamic equations (2.1), (2.7). For the system under study, a condensed phase appears if thefunction 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 ofphase transitions f = κ T n [ln( n / n a ) − + a n + b n + c n (2.16)where a , b , c are constant values. Three last terms in the formula (2.16) are the main terms. They arisedue to an exciton-exciton interaction and describe the phase transition. The fi rst term was introducedin order to describe the system in a space, where the exciton concentration is small (this is important ifsuch a region exists in a system). At an increase of the exciton density, the term an /2 manifests itself fi rstly. It contributes the an value to the chemical potential. In our system, the origin of this term isconnected with the dipole-dipole interaction, which should become apparent at the beginning with thegrowth of the density due to its long-range nature. To estimate a for the dipole-dipole exciton interactionin double quantum well we may use the plate capacitor formula an = π e dn / ǫ , where d is the distancebetween the wells, ǫ is the dielectric constant. This formula is usually used to determine the excitondensity from the experimental meaning of the blue shift of the frequency of the exciton emission withthe rise of the density. It follows from the formula that a = π e d / 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 thevalue b should be negative ( b < ). For stability of a system, at large n , the parameter c should be positive xciton condensation Figure 1.
The spatial dependence of the exciton density at a different value of the pumping: for a contin-ues line G = , for a periodical line G = , for a dashed line G = . D = , b = − . ( c > ). It is assumed in the model that the condensed phase arises due to the exchange and Van der Waalsinteractions. The calculations show that in some region of distances between the wells, these interactionsexceed the dipole-dipole repulsion.Let us introduce dimensionless parameters: ˜ n = n / n , where n = ( a / c ) , ˜ b = b /( ac ) , ˜ ~ r = ~ r / ξ ,where ξ = ( K / a ) is the coherence length, ˜ t = t / t , where t = κ T K /( Dn a ) , D = κ T /( an ) , ˜ G = Gt / n , ˜ τ ex = τ / t . As a result, the equation (2.15) is reduced to the form (hereinafter the symbol ∼ willbe omitted in the equation) ∂ n ∂ t = D ∆ n − n ∆ n + n ∆ n ¡ + bn + n ¢ − ~ ∇ n · ~ ∇ ∆ n + ¡ ~ ∇ n ¢ ¡ + bn + n ¢ + G − n τ ex . (2.17)The solutions of the equation (2.17) are presented in fi gure 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) = and the boundary conditions n ′ (0, t ) = n ′ ( L , t ) = n ′′ (0, t ) = n ′′ ( L , t ) = , where L is the size of a system. The periodical solution exists in someinterval of the pumping G c < G < G c . At speci fi ed parameters, the periodical solution exists at < G < . Outside this region, the solution describes a uniform system: the gas phase at a low pumpingand the condensed phase at a large pumping. The upper part of the periodical distribution correspondsto 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 G c to G c . Figure 2 shows the spatial dependence of theexciton current calculated by the formula (2.14). The current equals zero in the centers of the condensedand gas phases and it has a maximum in the region of a transition from the condensed phase to the gasphase. Let us do some estimations. The results for the currents in fi gure 2 are presented in dimensionlessunits: ˜ j = j / j , where j = n u , u = ( τ sc n a )/( m ξ ) is the unit of the velocity. The exciton density ispresented in fi gure 1 in dimensionless units ( ˜ n = n / n ). It is seen in fi gure 1 that ˜ n ∼ , and the magni-tude of n is of an order of n . Thus, for estimations we may assume that n a corresponds to the shift ofthe luminescence line with an increase of the exciton density, the magnitude of ξ is of the order of thesize of the condensed phase. For the following magnitudes of parameters τ sc = − s, n a = · − eV, m = · − g, ξ = · − cm, we obtain u ∼ cm/s. According to calculations (see fi gure 2), themagnitudes of the current and the velocity are two orders of magnitude less than their units j and u ,so the condition u ∼ cm/s takes place. In order to verify the ful fi lment of the condition (2.12), let ussuppose that ( ∂ u i )/( ∂ x k ∼ u / l ), where l is the period of a structure. It follows from experiments [4, 5] that l ∼ (5 ÷ µm. Using these data we see that the condition ( ) is very well satis fi ed. This condition isviolated at τ sc Ê − 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 hydrodynamicequation (2.15), the last term in equation (2.2)is of importance. It describes the loss of the momentum .I. Sugakov Figure 2.
The spatial dependence of the exciton current at G = , D = , b = − . due to the scattering of excitons by defects and phonons. It is this term that describes the processes thatcause a decay of the exciton fl ux. From the viewpoint of a possibility of the appearance of super fl uidity,the situation for excitons is more complicated than that for the liquid helium and for the atoms of alkalimetals 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 interactionbetween the atoms of a system and does not cause the change of the complete momentum of a systemand its movement as a whole. Phonons and defects for excitons are external subsystems that brake theexciton movement. Therefore, to create the exciton super fl uidity, it is needed that the value of τ sc shouldgrow signi fi cantly. 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 super fl uid transition is strongly lowered by inhomogeneities[36, 37]. Thus, the question regarding the possibility of the super fl uidity existence for indirect excitonson the basis of AlGaAs system is open.Thus, the peculiarities observed at large densities of indirect excitons may be explained by phasetransitions in a system of particles having attractive interactions and by the fi nite value of the lifetimewithout 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 regionbetween 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 interactionis impossible. This contradicts the main assumption of our works [19–25], though these works explainmany 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 con fi rmed 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 lowtemperature and at a small pumping, the main part of the emission band consists of the emission fromdefect 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 fromthe excitons (pairs of electrons and holes) localized on the defects. We assume that the localized statesare 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 localizedexcitations 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 ) = α N l exp( α E ) , where N l isthe density of the defect centers, E is the depth of the trap level. The exciton states (free and localized) are xciton condensation distributed onto levels after the creation of electrons and holes due to an external irradiation and theirsubsequent 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 thermodynamicalequilibrium state. In the considered model, we should obtain a distribution of electron-hole pairs, whosepopulation on a single level may be changed from zero to in fi nity for E > (for free exciton states) andfrom zero to one for E < (for localized states). Formally, in the considered system, free excitons haveBose-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 ofFermi-Dirac statistics for localized states on a single level for one trap is important. The equation forenergy distribution may be found from the minima of a large canonical distribution w ( n k , n i ) = exp µ Ω + N µ − E κ T ¶ , (3.1)where N = Σ i n i + Σ k n k , E = Σ i n i E i + Σ k , l n k E k , n i = , n k = ∞ , k is the wave vector of theexciton, 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 thefunctional (3.1). As a result, we obtain the following conditions for the mean values of the free excitondensity n and the density of localized states n L n ex = g ν π E ex a ex ∞ Z d E exp ³ E − µκ T ´ − (3.2) n L = α N l Z −∞ exp ( α E )d E exp ³ E − µκ T ´ + (3.3)where a ex = ( ħ ε )/( µ ex e ) and E ex = ( µ ex e )/(2 ε ħ ) are the radius and the energy of the exciton in theground state in the bulk material, g = , µ ex is the reduced mass of the exciton, ν is the ratio of the reducedand the total mass of the exciton. The chemical potential µ is determined from the condition n L + n = G τ ex , (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 = K , N l = a ex , α = eV ) − . .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 fi gure 4 (a) corresponds to the energy per a single exciton in the condensed phase. Onthe right [ fi gure 4 (b)], the upper line describes the whole emission from the island (the emission of boththe condensed phase and trapped excitons), the low line describes the emission of the trapped excitons. where G τ ex 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 fi gure 3as a function of the whole number of excitation presented in units of a ex . Let the exciton radius beequal to 10 nm. Then, the concentration of the traps and the width of the distribution of trap levels,chosen under calculations of fi gure 3, are of the order of cm − and 0.003 eV, correspondingly.As it is seen in fi gure 3, the number of localized excitations at small pumping exceeds the number offree excitons and the emission band is determined by the emission from the traps. With an increase ofpumping, the occupation of the trap levels becomes saturated. For the chosen parameters, the concentra-tion of excitations under saturation is of the order of cm − . The exciton density grows simultaneouslywith the saturation of the localized levels. As a result, the shortwave part of the emission band shouldincrease with an increased pumping. When the exciton density becomes larger, the collective excitoneffects begin to manifest themselves. The equations (3.2), (3.3) do not take into account the interactionsbetween excitations, and special models and theories are needed to describe the collective effects. Theappearance of a narrow line was observed in [6] with an increased pumping on the shortwave part of theexciton emission band. Simultaneously, patterns arise in the emission spectra. The narrow line appearedafter 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 thecondensed phase caused by the attractive interaction between excitons. The energy per a single excitonin the condensed phase is less than the energy of free excitons (the thick line in fi gure 4), but the gain ofthe 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 theislands of a condensed phase is higher than the energy of photons emitted by traps (see fi gure 4). Theexcitons cannot leave the condensed phase (the islands) and move to the traps (to the states of lowerenergy) since the levels of the traps are already occupied. This may be the reason of the results obtainedin [31], where the maximum of the frequency of emission from the islands is higher than the maximumfrequency 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 foranother 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 ofthe shortwave side of the band with an increased pumping.
4. Conclusion
Hydrodynamic equations are analyzed for excitons in a double quantum well. The equations takeinto account the presence of pumping, the fi nite value of the exciton lifetime and the possibility of acondensed phase formation in a phenomenological model. The equations describe the diffusion and theballistic movement of an exciton system. It is shown that the spatial nonuniform structures, observed xciton 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 theemission from condensed and gas phases is obtained. The theory explains the features of experimentaldependencies of the emission spectra from the condensed and gas phases. References
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В.Й. Сугаков
Iнститут ядерних дослiджень, просп. Науки, 47, 03680 Київ, УкраїнаПроведено аналiз рiвнянь гiдродинамiки екситонiв у квантовiй ямi. Рiвняння враховують 1) можливiстьфазового переходу в системi, 2) присутнiсть зовнiшньої накачки, 3) скiнчений час життя екситонiв, 4) роз-сiяння екситонiв на дефектах. Визначно порогову накачку утворення перiодичного розподiлу екситонноїгустини. Дослiджується вплив локалiзованих i вiльних екситонiв на формування спектрiв випромiнюван-ня.
Ключовi слова: самоорганiзацiя,квантовiями,екситони,фазовийперехiдсамоорганiзацiя,квантовiями,екситони,фазовийперехiд