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Amplification and passing through the barrier of the exciton condensed phase pulse in double quantum wells
Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Amplification and passing through the barrier of the exciton condensed phase pulse in double quantum wells O.I. Dmytruk a, V.I. Sugakov b,n a b
Taras Shevchenko National University of Kyiv, 2, Prosp. Academician Glushkov, Kyiv 03022, Ukraine Institute for Nuclear Research, National Academy of Sciences of Ukraine, 47, Prosp. Nauky, Kyiv 03680, Ukraine
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2013 Received in revised form 6 November 2013 Accepted 25 November 2013
The peculiarities and the possibility of a control of exciton condensed pulse movement in semiconductor double quantum wells under the slot in the metal electrode are studied. The condensed phase has been considered phenomenologically with the free energy in Landau–Ginzburg form taking into account the finite value of the exciton lifetime. It was shown that the exciton condensed phase pulse moves along the slot driven by an external linear potential. If the exciton density is high enough for the formation of the condensed phase then the pulse moves maintaining a constant value of a maximum density during exciton lifetime, while the exciton gas phase pulse diffuses. The penetration of the exciton condensed phase pulse through a barrier and its stopping by the barrier have been studied. Additionally, it was shown that the exciton pulse in the condensed phase can be amplified and recovered after damping by an additional laser pulse. Solutions for the system of excitons in double quantum wells under the slot in the electrode under steady irradiation in the form of bright and dark autosolitons were found. & 2013 Published by Elsevier B.V.
Keywords: Quantum wells Indirect excitons Condensed phase Pulse
1. Introduction In recent years, much attention has been paid to both experimental and theoretical study of indirect excitons in semiconductor double quantum wells at low temperatures. An indirect exciton is a bound pair of an electron and a hole which are separated by an electric field to different quantum wells [1]. Consequently, the recombination of the electron and the hole is inhibited, that causes the lifetime of indirect excitons to be several orders of magnitude higher than the direct exciton lifetime. The study of indirect excitons is promising in terms of fundamental science, because a high density of excitons can be created and many-exciton effects can be studied. Additionally, the system of indirect excitons can be promising for applications, since they can travel over large distances carrying energy and information and may be used in the double quantum well based semiconductor devices. Therefore, a great number of experimental and theoretical works have been devoted to the study of the indirect excitons' properties. The experimental studies on the AlGaAS-based quantum wells revealed [2–5] new nontrivial effects such as formation of spatially inhomogeneous structures (sometimes periodical) in the distribution of exciton density. Various spatial nonhomogeneous distributions of the exciton density were observed
n
Corresponding author. Tel.: þ 380 445254810; fax: þ380 445254463. E-mail addresses: [email protected] (O.I. Dmytruk), [email protected], [email protected] (V.I. Sugakov).
in the emission of the indirect excitons at the pumping greater than a certain critical value. Thus, in the paper [2] the authors observed a break-down of the emitting ring outside the laser spot into separate fragments periodically localized along the ring. In the papers [3,4], in which the excitation of the quantum well was carried out through a window in a metallic electrode, the authors found a periodical structure of the luminescent islands situated along the ring under the perimeter of the window. Recently Timofeev and coauthors [5] presented examples of the emitting structures, obtained through differently shaped windows in the electrodes: a rectangle, a triangle, two circles, two triangles etc. The appearance of the structures in the exciton density distribution was observed for a periodical potential applied to excitons [6]. Besides the periodical structure imposed by external conditions, the partition of the emission into fragments existed in the direction, in which the potential was almost uniform. The phenomenon of the break-down of the homogeneity and the emergence of emitting structures in systems of indirect excitons stimulated a number of theoretical investigations. Several different theoretical models of the formation of spatial patterns were proposed [7–14]. The authors of the work [7] considered the instability, which arises in the kinetics of the level occupation by particles with the Bose–Einstein statistics. Namely, the growth of the occupation of the level with zero momentum should stimulate the transitions of excitons to this level. But the required density of excitons was found to be greater, and the required temperature was found to be lower than the corresponding experimental values. Some authors explain the appearance of the periodicity
0921-4526/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physb.2013.11.055
Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i
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by the Bose condensation of excitons [8,9]. There is a suggestion to describe the system by a nonlinear Schrödinger equation [10]. The appearance of islands observed in Ref. [2] was explained in the paper [12] on the ground of a classical model of the diffusion with Coulomb interaction between the particles, but without taking into account the screening of the charges at macroscopic distances. The emergence of periodicity in the exciton system was investigated in Ref. [13] using Bogolyubov's equations with some approximations of inter-exciton interaction. Also a possibility of a Mott transition in considered systems has been studied [14]. In the listed works the main efforts were applied for the ascertainment of the principal possibility of the appearance of the periodicity in the exciton density distribution. A specific application of the results for the explanation of numerous experiments with different setups, at different pumping, temperature, different type of external fields was not employed. In the papers [15–19] we suggested a model which explained the experimentally observed spatial structures in the exciton density distribution. The theoretical approaches of the works [15–19] are based on the following assumptions: 1. There exists an exciton condensed phase caused by an attractive interaction between excitons. The assumption is supported by the existence of bound states of two indirect excitons, obtained in the calculations by several groups [20–22], and a new phase in the multiexciton system [23]. The attractive exchange and van der Waals interactions may exceed the repulsive long-range dipole–dipole interactions between excitons if the distance between the quantum wells in a double well setup, and, correspondingly, dipole moments of indirect excitons, are not too large. Note, that from the analysis of the shape of the exciton emission band, the authors of the work [24] came to the conclusion that the interaction between indirect excitons is exclusively repulsive. In the papers [25,26] an alternative explanation of the experimental result obtained in Ref. [24] was given, which does not contradict the assumption of an attractive interaction. It was shown that the important contribution to the formation of the exciton emission bandshape at low densities comes from the traps, levels of which are localized under the exciton band. These traps become saturated at increased pumping and, thus, the emission, coming from the band proper, shifts to a higher frequency. 2. The finite value of the exciton lifetime plays an important role in the formation of the exciton condensed phase. Usually the exciton lifetime is much greater than the time of establishment of the local equilibrium in a system. But taking into account the finiteness of the exciton lifetime is necessary in the study of the spatial distribution of phases in two-phase systems, because the exciton lifetime is less than the time of the establishment of the equilibrium between different phases. The latter is determined by slow diffusion processes and is large. The finite exciton lifetime restricts the maximal size of the condensed phase domain and causes the existence of a correlation in positions of separate regions of the condensed phase. The suggested model [15–19] has succeeded in explaining the spatial structures observed in the luminescence of indirect excitons. Particularly, this model has allowed us to describe the spatial distribution of the excitons on the ring outside the laser spot, observed in Ref. [2], the spatial structure of the luminescence under the window in the metal electrode, observed in Refs. [3,4]. The involvement of Bose–Einstein condensation for excitons was not required for the explanation of the experiments. The Bose– Einstein nature of excitons contributes to the values of the phenomenological coefficients, which may be obtained in microscopic theory. But the condensation occurs in a real space due to
the interaction between excitons. The application of the presented model of the exciton condensation in the model of nucleation [15,17] is similar to a study of creation of electron–hole drops in semiconductors [27–29]. In this case, the islands of the exciton condensed phase are a two dimensional analog of the electron–hole drops. But, compared to the works [27–29], we took into account the correlation of the mutual positions of the islands, because they draw excitons from the same environment. This correlation allows us to explain the emergence of the periodicity in the distribution of the islands, which is observed in experiments. The possibility of using the system of indirect excitons in the optoelectronics is studied in several works [30–34]. The experimental possibility of building an exciton optoelectronic transistor [30], an excitonic integrated circuit [31] and an excitonic conveyor [32] was demonstrated recently. Therefore, the theoretical study of the propagation of the interacting system of indirect excitons in external fields is important. In this work we study the formation, amplification and passing through the barrier of the exciton condensed phase pulse in the semiconductor double quantum well in the setup, in which one of the electrodes has a slot, through which the excitons are created. We consider this problem using our model [15–19], approbated by the analysis of experimental results. The formation of the exciton condensed phase in the double quantum well under the slot excited by a steady uniform irradiation was studied in Ref. [19]. Unlike the work [19] this paper investigates the excitation of the exciton condensed phase by a pulse irradiation. This excitation creates an exciton condensed phase pulse, the movement and control of which are analyzed. The possibility of the amplification of a pulse by an additional laser pulse in the one-dimensional system was investigated in Ref. [35]. Compared to Ref. [35], in the presented paper, the processes of passing of the exciton pulse through a barrier and the possibility of stopping of the pulse by the barrier are investigated. Also we consider the real physical two-dimensional system (a slot in the electrode) and estimate the effect of exciton–exciton annihilation on the traveling pulse. The analysis of the difference in the movement of the pulses built from the excitons in a gas phase and the excitons in a condensed phase is presented.
2. Model of the system We consider the semiconductor structure with the double quantum well sandwiched between two metal electrodes. There is the slot in the upper electrode (Fig. 1). The width of the slot is 2b. Let us choose the OX axis along the slot, the axis OY in the transverse direction and the OZ axis along the normal to the electrode. Let the plane of the quantum well has a coordinate z ðz o 0Þ. The presence of the slot forms an inhomogeneous electric field that is additional to the homogeneous field of the solid electrode and gives an additional potential energy for excitons in quantum wells. The distribution of the exciton density in the double quantum well was determined [19] for the uniform steady-state excitation of the well through the slot. It was shown that a chain of periodically situated islands arises in the well below the center of the slot if the pumping exceeds some critical value. If the width of the slot is large, two chains of exciton condensed phase islands, located along the edges of the slot, appear. Additionally, the possibility of a movement of the chains along the slot was shown in Ref. [19] if driven by an external potential. In the considered system presented in Fig. 1, there are additional elements in comparison to the system investigated in the work [19]. We shall see, that these elements allow controlling the propagation of the exciton pulse.
Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i
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But in a certain vicinity of the borders of the slot the potential has a small minimum with a negative value. It appears due to the rearrangement of charges on the conductive electrode at the edges. The depth of the minimum increases if the quantum well is situated closer to the electrode with the slot.
3. Method of solution
Fig. 1. Setup of the system. The semiconductor double well structure is sandwiched between two metal electrodes. There is a slot in the upper electrode. U1 and U2 are the potentials on the upper and lower electrodes, correspondingly. Ug is the gate potential of the strip that creates a barrier for excitons. Arrows indicate the laser irradiation.
Fig. 2. Dependence of the potential created by the slot on the distance from the center of the slot in the plane of the quantum well for V 0 ¼ 5, z ¼ 15 and for different width of the slot: (1) b ¼ 10 (dashed curve), (2) b ¼7 (dotted curve), (3) b ¼4 (solid curve). The parameters are shown in dimensionless units, which will be introduced in the next section.
In the case of the strong electric field, created by electrodes, the potential energy Vtot for exciton in the quantum well, may be presented as V tot ¼ pz Eðy; zÞ, where pz is the exciton dipole moment, Eðy; zÞ is the electric field strength in the y; z that are point of the quantum well. This electric field is uniform in the region far from the slot Eðy; zÞ ¼ E0 . Additional potential for excitons, created by slot, is equal to V ðy; zÞ ¼ V tot V 0 , where V 0 ¼ pz E0 is the uniform shift of the exciton band by the electric field in the region distant from the slot. Using the solution of the problem for the electric field of a grounded conducting plane with a slot in an external electric field [36], and making the assumption that the distance between the electrodes is much greater than the width of the slot, we obtain the following equation for the slot contribution into the potential energy of an exciton: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V0 b z2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vðy; zÞ ¼ ð 1 þb =ξðy; zÞ 1Þ 2 2 2ξðy; zÞ2 1 þ b =ξðy; zÞ 0 13 2
y2 þ z2 þ b B C7 @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5; 2 2 2 2 2 2 ðy þ z b Þ þ4z b where
ð1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ξðy; zÞ ¼ 12 ½y2 þ z2 b2 þ ðy2 þz2 b2 Þ2 þ 4b2 z2 :
ð2Þ
The potential, created by the slot for exciton in the quantum well, is positive under the center of the slot (Fig. 2), because the electric field in this regions is less than the field far from the slot.
In this section we will concentrate our attention on the main equations for finding the excitonic density dependence on time and the position in space. In order to study the exciton distribution, let us consider the conservation law for the exciton density: ! ∂n n ! ¼ div j þ Gð r ; tÞ ; ∂t τ
ð3Þ
! where n is the exciton density, Gð r ; tÞ is the pumping (the number of excitons created in a unit time in a unit area of the quantum ! ! well), τ is the exciton lifetime, j ¼ M ∇ μ is the density of the exciton current, M is the exciton mobility. We use the Einstein formula M ¼ nD=kT for the exciton mobility, where D is a diffusion coefficient, k is the Boltzmann constant, T is the temperature. The described system is non-equilibrium due to the creation and decay of excitons. The system is also inhomogeneous. The assumption of the establishment of local equilibrium allows description of the system by a free energy depending on the exciton density, in its turn depending on the spatial coordinates. In such a way the free energy depends on the coordinates. Such an assumption is typical and important for the study of the non-equilibrium systems [37]. Having expressed the chemical potential as μ ¼ δF=δn, we chose the free energy in the Landau form: Z ! K ! 2 ð ∇ nÞ þ f ðnÞ þ nV tot : F½n ¼ dr ð4Þ 2 ! The term K=2ð ∇ nÞ2 characterizes the energy of inhomogeneity, f (n) is a free energy density of a uniform system, nVtot accounts for the potential energy in the external field. We approximate the density of the free energy by the following: n a b c ð5Þ f ðnÞ ¼ f 0 þ kTn ln 1 þ n2 þ n3 þ n4 ; n0 2 3 4 where the terms of the power series for the exciton density are important for large n and for the description of the condensed phase, while the term kTnðln n=n0 1Þ may be essential only if one wants to describe a system in which the density of excitons is small and the interaction between them is not important. Introducing dimensionless variables for length, density, energy and time: rffiffiffiffi rffiffiffi 2 K a d1 l u lu ¼ ; V u ¼ anu ; t u ¼ ; ð6Þ ; nu ¼ a c D and substituting Eq. (4) into Eq. (3), we obtain a nonlinear equation that governs the exciton density: ∂n ! ! ¼ d1 Δn þ ∇ ½n ∇ ð Δn þn ∂t n þb1 n2 þ n3 þV Þ þG gn2 ;
τ
ð7Þ
pffiffiffiffiffi where b1 ¼ b= ac o0, d1 ¼ kT=V u , and V is the potential created by the slot and given by Eq. (1). Let us do analysis of an influence of the external potential on the free energy. Since the phenomenological coefficients (a,b,c) in Eq. (5) describe the interaction between excitons, they should depend on the orientation of dipoles due to the dipole–dipole interaction. In order to create indirect excitons in double wells (to separate the charges) the strong electric field should be applied between electrodes. At low temperature, the strong external
Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i
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electric field orients exciton dipoles perpendicular to the wells' plane independently on the phase state of excitons and the screening does not occur (dipoles are already oriented entirely). The potential V, created but the slot, is much smaller than the potential of uniform electric field far from the slot V0 ðjVj⪡jV 0 jÞ. So, it does not affect the dipole orientation, and, therefore, the coefficients of the free energy (a,b,c), which are the same anywhere in the quantum well. The exciton–exciton annihilation (Auger process) was taken into account by introducing the term gn2 in Eq. (7). Eq. (7) is a non-linear phenomenological equation that describes the distribution of exciton density taking into account ! the pumping Gð r ; tÞ and the finite lifetime τ.
Fig. 3. Time evolution of the exciton density maximum for different sets of parameters. Squares correspond to the exciton pulse in the condensed phase ðP ¼ 0:64; g ¼ 0Þ, triangles correspond to the exciton pulse in the gas phase ðP ¼ 0:1; g ¼ 0Þ, circles correspond to the exciton pulse in the condensed phase taking into account Auger recombination of excitons ðP ¼ 0:64; g ¼ 7 10 4 Þ, stars correspond to the exciton pulse in the gas phase taking into account Auger recombination of the excitons ðP ¼ 0:1; g ¼ 7 10 4 Þ. b ¼ 4, β ¼ 10, Δt ¼ 20, τ ¼ 1000, b1 ¼ 2:23, d1 ¼ 0.2.
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4. Creation and movement of the exciton condensed phase pulses A domain in which the exciton density is high can be formed by illuminating the system with a laser pulse. We assume the excitons are generated during a time period Δt with the position dependent rate: ! 8 2 2 > < P exp ðx x0 Þ þ ðy y0 Þ if t t 1 o Δt; jy y0 j o b; Gðx; y; tÞ ¼ β2 > : 0 in all other cases: ð8Þ Eq. (8) takes into account the non-transparency of the electrodes. P is proportional to the number of excitons created by the laser per unit time in a unit area of the quantum well plane. The laser pulse causes heating. Considering this problem, we assume that the time by which the temperature reaches equilibrium is much less than the exciton lifetime τ. The exciton density in the domain may be sufficient for the formation of the condensed phase. We will refer to such domains are an exciton condensed phase pulse. In the opposite case of the exciton density insufficient for the condensation the domain will be referred to as the exciton gas pulse. Time evolution of two types of pulses is different. This difference can be illustrated by considering the evolution of the exciton density with time for standing pulses, id est when there is no force driving excitons to move along the slot. First, we consider the decay of the pulse neglecting the exciton–exciton annihilation (g ¼0). The dependence of the peak values of exciton density in the pulses is presented in Fig. 3 for two magnitudes of the laser pulses: P¼ 0.1, g¼ 0 and P ¼ 0:64; g ¼ 0. If the value of the amplitude of the laser pumping (P ¼0.1) is less than the threshold value, the exciton pulse is formed in the gas phase. It is seen from Fig. 3 that in this case the peak of the exciton density in the pulse decreases rapidly with time after the generation stops, and the width of the exciton pulse increases. If the value of the amplitude of the generation pulse exceeds the threshold value (P¼0.64), the excitons in the pulse form the condensed phase. The peak value of the exciton density in
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Fig. 4. Spatial distribution of the exciton density at b¼ 4, P¼ 0.64, β ¼ 10, τ ¼ 1000, b1 ¼ 2:23, d1 ¼ 0.2, δ ¼ 0:09 at different moments of time after switching off the pumping: (a) t ¼ 100, (b) t¼ 500, (c) t ¼900, (d) t ¼1300. The white lines show the edges of the slot.
Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i
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this case remains approximately constant during the exciton lifetime. Therefore, the attractive interaction between excitons in the condensed phase inhibits the spread-out of the domain. As the excitons decay with time, the width of the pulse decreases, it thins down, but the density maximum preserves the same value over long time till the system is found in the condensed phase state. To study the effect of the annihilation, we have chosen a large value of the annihilation rate, such that the lifetimes due to the annihilation and radiation are the same. It is seen from Fig. 3 that annihilation affects two types of pulses in the qualitatively different ways. For the condensed phase pulse ðP ¼ 0:64; g ¼ 0:0007Þ, the duration of the period, in which the maximum of the exciton density remains constant, decreases compared to the case when annihilation is neglected ðP ¼ 0:64; g ¼ 0Þ. The gas phase pulse is affected by the annihilation weakly as seen from comparison of the time dependence of the density profiles for P ¼ 0:1; g ¼ 0:0007 and P ¼ 0:1; g ¼ 0. Such behavior is obviously caused by the fact that the exciton density in the condensed phase is larger. The next problem we have considered is the traveling of the pulse. We assume that the excitons are driven along the slot by a bias represented in the free energy by an additional potential V l ðxÞ ¼ δx;
ð9Þ
where δ is a constant. Such potential can be created, for example, by using an electric field, as it the authors [38] did, or by applying an inhomogeneous stress to the crystal. Four snapshots of the exciton pulse taken at different moments of time are shown in Fig. 4. Additionally, Fig. 5 plots the projections of density distribution onto the XZ plane. One can see that the exciton condensed phase pulse moves along the slot thinning down with time as the excitons decay. The velocity of the propagation depends on the potential ðv δD=kTÞ. Provided the exciton pulse is in the condensed phase, the maximum value of the exciton density in the pulse remains almost constant during its lifetime, while the width of the pulse decreases. In the case of the gas phase pulse, the profile spreads out. Two exciton pulses are formed in the case of a wide slot. They also move along the slot parallel to each other. Exciton pulses can be used for transmitting information. The condensed phase pulses may transfer signal to greater distances, if the detector at the output reacts to the amplitude of the exciton density.
Fig. 5. Dependence of the exciton density on the x coordinate at different moments of time in the system without the barrier (parameters are the same as in Fig. 4).
5
5. Amplification of the exciton condensed phase pulse by a laser pulse An exciton condensed phase pulse can be amplified by an additional laser pulse. We have shown already that the exciton density within the condensed phase remains constant during the exciton lifetime, but at a certain moment of time the amplitude of the pulse begins to drop down and the system of excitons in the pulse transforms to the gas phase. To recover the exciton condensed phase pulse, let us shine an additional laser pulse at a certain fixed moment of time. The amplitude of this additional pulse may be lower than the threshold value for the creation of the condensed phase pulse. Matching the position of the additional laser pulse with the position of the traveling exciton pulse leads to the transition of the exciton pulse to the condensed phase. Thus, two laser pulses in a sequence help to maintain the condensed phase, the exciton pulse exists for a longer period of time and can travel further. The time dependence of the maximum values of the exciton density in the first, the second and resulting pulses is plotted in Fig. 6. If the detector reacts to the amplitude of the exciton pulse, then the signal carried by excitons in the condensed phase can be detected at greater distances.
6. Control of the exciton condensed phase pulse using the barrier The propagation of the exciton condensed phase pulse can be controlled, for example, using a barrier. The barrier can be created by placing a thin metal strip perpendicularly to the slot (see Fig. 1). The barriers of different magnitudes can be formed by applying voltage to the strip. Let us approximate the additional potential for excitons created by the strip as follows: V b ðxÞ ¼ S expð ðx x0 Þ2 =2ðΔxÞ2 Þ;
ð10Þ
where S is the magnitude of the barrier and Δx is the width of the barrier. The cumulative potential for excitons created by the linear potential Eq. (9) and the barrier Eq. (10) is shown in Fig. 7, the parameters of these potentials are δ ¼ 0:09, S¼ 1. One can observe that in some cases (S ¼1) an obstacle is created on the way of the traveling exciton condensed phase pulse. Let us analyze the effect of the height of the barrier on the penetration of the pulse through the barrier. The comparison of the projections of the exciton density onto the YZ plane for S ¼0
Fig. 6. Dependence of the exciton density maximum (nmax) on time (t). Squares correspond to the first exciton pulse (P¼ 0.64), triangles correspond to the second pulse (P¼ 0.1), circles correspond to the total pulse and the dashed line corresponds to the maximum of the decreasing exciton density without additional laser pulse.
Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i
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Fig. 7. Total additional potential for excitons created by linear potential and barrier ðV l ðxÞ þ V b ðxÞÞ, δ ¼ 0:09, S ¼1.
Fig. 8. Dependence of the exciton density on the x coordinate at different moments of time in the presence of the barrier at S ¼0.2, Δx ¼ 2:24, δ ¼ 0:09, b ¼ 4, P ¼0.64, β ¼ 10, Δt ¼ 20, τ ¼ 1000, b1 ¼ 2:23, d1 ¼0.2.
Fig. 9. Dependence of the exciton density on the x coordinate at different moments of time in the presence of the barrier, S ¼0.3, Δx ¼ 2:24, δ ¼ 0:09 (other parameters are the same as in Fig. 8).
(Fig. 5) and S¼ 0.2 (Fig. 8) shows that the amplitude of the pulse after passing the barrier drops down with increasing the height of the barrier. For large S the exciton condensed phase pulse does not survive passing the barrier (Fig. 9), the pulse in the condensed phase exists beyond the barrier only at S r0:3. The dependence of
the exciton density on spatial coordinates in the case of the barrier with parameters S ¼1, Δx ¼ 2:24 and the position of the maximum x0 ¼ 100 is presented as the two-dimensional plot in Fig. 10 and as the projection onto the YZ plane in Fig. 11. The position of the pulse does not change with time after the pulse reaches the barrier. The exciton condensed phase pulse stops in the area before the barrier. This demonstrates the possibility of controlling the motion of the pulse by changing the barrier parameters. As the pulse stops, it changes the shape (see Fig. 10). The collective penetration of excitons through a barrier is somewhat similar to the tunneling of a quantum-mechanical particle through a barrier: the probability of the tunneling decreases for higher barriers as well. The dependence on the width of the barrier (on Δx) is not straightforward, since we considered the potential for excitons as consisting of two parts: a linear part of the driving force and the potential of the barrier. Depending on the values of Δx and δ the total height of the barrier may either decrease or increase as a function of Δx (see Fig. 7) with different effects on the pulse penetration. We have carried out our calculations in dimensionless units. Let us perform some estimates to represent the results in real dimensional quantities. We chose the following values of parameters: τ ¼ 100 ns, T ¼3.7 K, nu ¼ 2 1010 cm 2 , D¼9 cm2/s, anu ¼1.6 meV. In this case the width of the slot 2b is 5:36 μm (b¼4). The traveling path of the exciton condensed phase pulse during its lifetime is l ¼ 67 μm, and the height of the barrier stopping the pulse is S ¼ anu ¼ 1:6 meV (S ¼1). The narrowing of the pulse with time is indicative of the attractive interaction between the particles forming the pulse. This process differs from the Bose–Einstein condensation, in which the narrowing of the distribution occurs in the momentum space, that should lead to a broadening of the spatial width. Therefore, an experimental study of the behavior of the domain may determine the nature of the condensed phase.
7. Exciton autosolitons So far we have studied the exciton density pulses localized in space. These pulses are created by laser pulses acting during a certain time period. Exciton density pulses created this way gradually weaken and finally disappear. However, the localized in space exciton pulses (autosolitons) are possible at stationary excitation by irradiation in a certain interval of the exciton generation rate as it was shown for the one-dimensional system [25,26]. Let us consider the possibility of the formation of the exciton autosolitons in the potential created by the electrode with the slot. As shown in Ref. [19], the periodical distribution of the exciton density in the quantum well emerges for the stationary irradiation below the slot in the electrode in a certain interval of the pumping Gc1 o G oGc2 . The homogeneous distribution of the exciton density exists outside this interval of the exciton generation rate. We showed that in this case of two-dimensional system at steady-state pumping there is a spatially nonuniform solution of Eq. (7) for G oGc1 in the form of an isolated peak as in one dimensional system [25,26]. This solution may be obtained by solving Eq. (7) with the generation rate consisting of a constant value G0 and an additional pulse dG with a maximum at a certain point in space and at the time moment dG ¼ s exp½ wððx x0 Þ2 þ ðy y0 Þ2 Þ exp½ pðt t 0 Þ2 ;
ð11Þ
where s; w; p are parameters. Eq. (11) describes a pulse of the pumping acting during a period of time and having the maximum at the point x0 ; y0 .
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Fig. 10. Spatial distribution of the exciton density for S ¼1, Δx ¼ 2:24, δ ¼ 0:09 (other parameters are the same as in Fig. 8) at different moments of time after switching off the pumping: (a) t¼ 100, (b) t¼ 500, (c) t ¼900, (d) t ¼1300. The white lines show the edges of the slot.
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Fig. 12. Bright autosoliton. Spatial distribution of the exciton density for G ¼ 0:005 o Gc1 , b ¼7, b1 ¼ 2:23, x0 ¼ 60, y0 ¼ 30. Fig. 11. Dependence of the exciton density on the x coordinate at different moments of time in the presence of the barrier, S¼ 1, Δx ¼ 2:24, δ ¼ 0:09 (other parameters are the same as in Fig. 8).
The solution of Eq. (7) for the system of indirect excitons under the slot in the metal electrode obtained in the case of imposing the addition pulse Eq. (11) is presented in Fig. 12. The solution exists at t-1, i.e. at the times, when the action of the addition pulse is already absent. The shape of the peak nðx; yÞ does not depend on parameters s; w; p in some region of their values, but this solution exists in a finite interval of the pumping ðG ¼ 0:0048 0:0052Þ. In addition, the solution in the form, presented in Fig. 12, arises also, if the additional pulse is absent, but there is distribution of the exciton density in the initial moment of time in the form: nðx; y; 0Þ ¼ s expð wððx x0 Þ2 þ ðy y0 Þ2 ÞÞ:
ð12Þ
Such localized spatial distribution of the exciton density is called “bright autosoliton”. According to the classification of Ref. [39] it belongs to the class of static soliton. Localized solutions exist also in a certain region of pumping greater the value, at which the periodical structure arises ðG 4 Gc2 Þ. The dependence of the exciton density may be obtained from Eq. (7) choosing an additional pumping pulse in the form Eq. (11),
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Fig. 13. Dark autosoliton. Spatial distribution of the exciton density for G ¼ 0:008 4 Gc2 , b ¼7, b1 ¼ 2:23, x0 ¼ 60, y0 ¼ 30.
but with s o 0. Such solution is presented in Fig. 13. These structures can be called “dark autosolitons”. They exist in a certain region of pumping ðG ¼ 0:0078 0:0081Þ. If there is an external field in the system creating a linear potential (see Eq. (9)), the autosoliton moves along the slot. So, these autosolitons can be
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used for transmitting information as well. In this case, the motion occurs at steady-state irradiation, but such pulse does not decay with time. 8. Conclusions We have considered the dynamics of the exciton condensed phase pulse, created in double quantum wells by the external irradiation. The modeled system consists of a double well structure sandwiched between two metal electrodes, the top one having a slot that creates an additional potential for excitons. The spinodal decomposition approach, generalized to the case of particles with a finite lifetime, was used to describe the evolution of the exciton density distribution in condensed and gas phases. To ensure traveling of the exciton pulse, an additional potential linearly depending on the coordinate along the slot was included. The possibility to control the propagation of the exciton pulse was demonstrated using either the barrier, which arises due to the potential created by a metal strip, or the additional laser pulse. The numerical simulations produced the following results: 1. Exciton condensed phase pulse arise if the intensity of the laser pulse is greater than the threshold value dependent on the parameters of the system. 2. In the case of a narrow slot, an exciton condensed phase pulse is formed in the center of the slot, and in the case of a wide slot, the exciton condensed phase pulse splits into two pulses, located near the boundaries of the slot. 3. In the presence of the linear driving potential along the slot, the exciton pulse moves with a constant velocity. Maximum density remains constant during the exciton lifetime and the pulse does not spread out if the exciton system is in the condensed phase. The width of the pulse decreases with the time. 4. The attenuated pulse can be amplified by imposing an additional laser pulse thereby increasing the path traversed before the disappearance. 5. The exciton pulse can be stopped or its shape can be changed by creating a potential barrier for excitons. With increasing the height of the barrier, the permeation of the excitons through the barrier decreases and the shape of the pulse changes stronger. 6. The existence of the bright and dark autosolitons in the system of excitons in double quantum wells under the slot in the electrode at steady pumping was demonstrated. The paper analyses how the exciton motion in inhomogeneous external fields is influenced by the collective effects leading to the formation of the condensed phase. The experimental observation of the pulse behavior will allow us to determine the origin of the condensed phase. If the condensed phase is caused by attractive interaction between excitons, the spatial width of the pulse should decrease during the existence of the condensed phase. In the case of the Bose–Einstein condensation the narrowing of the pulse should occur in momentum space, that leads to a broadening of
the spatial width. Also the spatial increase of the width of the pulse must occur for the pulse of excitons in the gas phase. Moreover, the investigations of the exciton pulses may be perspective for application of the exciton systems in the optoelectronics.
57 58 59 60 61 62 Acknowledgments 63 64 65 The work was supported partially by the joint grant F53.2/033 66 of the State Fund for Fundamental Research and Russian Founda67 tion for Basic Research. The authors thank to Dr. I. Yu. Goliney for 68 useful discussions. 69 70 References 71 72 [1] T. Fukuzawa, E.E. Mendez, J.M. Hong, Phys. Rev. Lett. 64 (25) (1990) 3066. 73 [2] L.V. Butov, A.C. Gossard, D.S. Chemla, Nature 418 (6899) (2002) 751. 74 [3] A.V. Gorbunov, V.B. Timofeev, JETP Lett. 84 (6) (2006) 329. [4] A.V. Gorbunov, V.B. Timofeev, Phys.-Uspekhi 49 (6) (2006) 629. 75 [5] V.B. Timofeev, A.V. Gorbunov, D.A. Demin, Low Temp. Phys. 37 (2011) 179. 76 [6] M. Remeika, J.C. Graves, A.T. Hammark, et al., Phys. Rev. Lett. 102 (2009) 77 186803. [7] L.S. Levitov, B.D. Simons, L.V. Butov, Phys. Rev. Lett. 94 (17) (2005) 176404. 78 [8] A.V. Paraskevov, T.V. Khabarova, Phys. Lett. A 368 (1) (2007) 151. 79 [9] R.B. Saptsov, JETP Lett. 86 (10) (2008) 687. 80 [10] C.S. Liu, H.G. Luo, W.C. Wu, J. Phys.: Condens. Matter 18 (42) (2006) 9659. [11] V.K. Mukhomorov, Phys. Solid State 52 (2) (2010) 241. 81 [12] J. Wilkes, E.A. Muljarov, A.L. Ivanov, Phys. Rev. Lett. 109 (18) (2012) 187402. 82 [13] S.V. Andreev, arXiv, arXiv:1212.2748, in press. Q2 83 [14] V.S. Babichenko, I.Ya. Polishchuk, Pis'ma Zh. Ekp. Theor. Fiz 97 (11–12) (2013). [15] V.I. Sugakov, Solid State Commun. 134 (1) (2005) 63. 84 [16] A.A. Chernyuk, V.I. Sugakov, Phys. Rev. B 74 (8) (2006) 085303. 85 [17] V.I. Sugakov, Phys. Rev. B 76 (11) (2007) 115303. 86 [18] V.I. Sugakov, A.A. Chernyuk, JETP Lett. 85 (11) (2007) 570. [19] V.I. Sugakov, J. Phys.: Condens. Matter 21 (27) (2009) 275803. 87 [20] M.Y.J. Tan, N.D. Drummond, R.J. Needs, Phys. Rev. B 71 (3) (2005) 033303. 88 [21] C. Schindler, R. Zimmermann, Phys. Rev. B 78 (4) (2008) 045313. 89 [22] A.D. Meyertholen, M.M. Fogler, Phys. Rev. B 78 (23) (2008) 235307. 90 [23] Yu E. Lozovik, O.L. Berman, J. Exp. Theor. Phys. Lett. 64 (8) (1996) 573. [24] Sen Yang, A.V. Mintsev, A.T. Hammack, L.V. Butov, A.C. Gossard, Phys. Rev. B 75 91 (3) (2007) 033311. 92 [25] V.I. Sugakov, Ukr. J. Phys. 56 (10) (2011) 1124. 93 [26] V.I. Sugakov, arXiv, arXiv:1306.4876, in press. [27] R.N. Silver, Phys. Rev. B 11 (1976) 1569. 94 [28] R.M. Westervelt, Physica Status Solidi 74 (1976) 727. 95 [29] V.S. Bagaev, N.V. Zamkovets, L.V. Keldysh, N.N. Sybel'din, V.A. Tsvetkov, Sov. 96 Phys. JETP 43 (1976) 783. [30] A.A. High, A.T. Hammack, L.V. Butov, M. Hanson, A.C. Gossard, Opt. Lett. 32 (17) 97 (2007) 2466. 98 [31] A.A. High, E.E. Novitskaya, L.V. Butov, M. Hanson, A.C. Gossard, Science 321 99 (5886) (2008) 229. [32] A.G. Winbow, J.R. Leonard, M. Remeika, Y.Y. Kuznetsova, A.A. High, 100 A.T. Hammack, L.V. Butov, J. Wilkes, A.A. Guenther, A.L. Ivanov, et al., Phys. 101 Rev. Lett. 106 (19) (2011) 196806. 102 [33] M. Remeika, J.C. Graves, A.T. Hammack, A.D. Meyertholen, M.M. Fogler, 103 L.V. Butov, M. Hanson, A.C. Gossard, Phys. Rev. Lett. 102 (18) (2009) 186803. [34] Y.Y. Kuznetsova, M. Remeika, A.A. High, A.T. Hammack, L.V. Butov, M. Hanson, 104 A.C. Gossard, Opt. Lett. 35 (10) (2010) 1587. 105 [35] O.I. Dmytruk, V.I. Sugakov, Phys. Lett. A 376 (44) (2012) 2804. 106 [36] L.D. Landau, Electrodynamics of Continuous Media: Landau and Lifshitz Course of Theoretical Physics, vol. 8, 1998. Q3107 [37] P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability and 108 Fluctuations, 1971. 109 [38] J.R. Leonard, M. Remeika, M.K. Chu, Y.Y. Kuznetsova, A.A. High, L.V. Butov, J. Wilkes, M. Hanson, C. Gossard, Appl. Phys. Lett. 100 (23) (2012) 231106. 110 [39] B.S. Kerner, V.V. Osipov, Phys.-Uspekhi 32 (2) (1989) 101. 111
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Amplification and passing through the barrier of the exciton condensed phase pulse in double quantum wells O.I. Dmytruk a, V.I. Sugakov b,n a b
Taras Shevchenko National University of Kyiv, 2, Prosp. Academician Glushkov, Kyiv 03022, Ukraine Institute for Nuclear Research, National Academy of Sciences of Ukraine, 47, Prosp. Nauky, Kyiv 03680, Ukraine
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2013 Received in revised form 6 November 2013 Accepted 25 November 2013
The peculiarities and the possibility of a control of exciton condensed pulse movement in semiconductor double quantum wells under the slot in the metal electrode are studied. The condensed phase has been considered phenomenologically with the free energy in Landau–Ginzburg form taking into account the finite value of the exciton lifetime. It was shown that the exciton condensed phase pulse moves along the slot driven by an external linear potential. If the exciton density is high enough for the formation of the condensed phase then the pulse moves maintaining a constant value of a maximum density during exciton lifetime, while the exciton gas phase pulse diffuses. The penetration of the exciton condensed phase pulse through a barrier and its stopping by the barrier have been studied. Additionally, it was shown that the exciton pulse in the condensed phase can be amplified and recovered after damping by an additional laser pulse. Solutions for the system of excitons in double quantum wells under the slot in the electrode under steady irradiation in the form of bright and dark autosolitons were found. & 2013 Published by Elsevier B.V.
Keywords: Quantum wells Indirect excitons Condensed phase Pulse
1. Introduction In recent years, much attention has been paid to both experimental and theoretical study of indirect excitons in semiconductor double quantum wells at low temperatures. An indirect exciton is a bound pair of an electron and a hole which are separated by an electric field to different quantum wells [1]. Consequently, the recombination of the electron and the hole is inhibited, that causes the lifetime of indirect excitons to be several orders of magnitude higher than the direct exciton lifetime. The study of indirect excitons is promising in terms of fundamental science, because a high density of excitons can be created and many-exciton effects can be studied. Additionally, the system of indirect excitons can be promising for applications, since they can travel over large distances carrying energy and information and may be used in the double quantum well based semiconductor devices. Therefore, a great number of experimental and theoretical works have been devoted to the study of the indirect excitons' properties. The experimental studies on the AlGaAS-based quantum wells revealed [2–5] new nontrivial effects such as formation of spatially inhomogeneous structures (sometimes periodical) in the distribution of exciton density. Various spatial nonhomogeneous distributions of the exciton density were observed
n
Corresponding author. Tel.: þ 380 445254810; fax: þ380 445254463. E-mail addresses: [email protected] (O.I. Dmytruk), [email protected], [email protected] (V.I. Sugakov).
in the emission of the indirect excitons at the pumping greater than a certain critical value. Thus, in the paper [2] the authors observed a break-down of the emitting ring outside the laser spot into separate fragments periodically localized along the ring. In the papers [3,4], in which the excitation of the quantum well was carried out through a window in a metallic electrode, the authors found a periodical structure of the luminescent islands situated along the ring under the perimeter of the window. Recently Timofeev and coauthors [5] presented examples of the emitting structures, obtained through differently shaped windows in the electrodes: a rectangle, a triangle, two circles, two triangles etc. The appearance of the structures in the exciton density distribution was observed for a periodical potential applied to excitons [6]. Besides the periodical structure imposed by external conditions, the partition of the emission into fragments existed in the direction, in which the potential was almost uniform. The phenomenon of the break-down of the homogeneity and the emergence of emitting structures in systems of indirect excitons stimulated a number of theoretical investigations. Several different theoretical models of the formation of spatial patterns were proposed [7–14]. The authors of the work [7] considered the instability, which arises in the kinetics of the level occupation by particles with the Bose–Einstein statistics. Namely, the growth of the occupation of the level with zero momentum should stimulate the transitions of excitons to this level. But the required density of excitons was found to be greater, and the required temperature was found to be lower than the corresponding experimental values. Some authors explain the appearance of the periodicity
0921-4526/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physb.2013.11.055
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by the Bose condensation of excitons [8,9]. There is a suggestion to describe the system by a nonlinear Schrödinger equation [10]. The appearance of islands observed in Ref. [2] was explained in the paper [12] on the ground of a classical model of the diffusion with Coulomb interaction between the particles, but without taking into account the screening of the charges at macroscopic distances. The emergence of periodicity in the exciton system was investigated in Ref. [13] using Bogolyubov's equations with some approximations of inter-exciton interaction. Also a possibility of a Mott transition in considered systems has been studied [14]. In the listed works the main efforts were applied for the ascertainment of the principal possibility of the appearance of the periodicity in the exciton density distribution. A specific application of the results for the explanation of numerous experiments with different setups, at different pumping, temperature, different type of external fields was not employed. In the papers [15–19] we suggested a model which explained the experimentally observed spatial structures in the exciton density distribution. The theoretical approaches of the works [15–19] are based on the following assumptions: 1. There exists an exciton condensed phase caused by an attractive interaction between excitons. The assumption is supported by the existence of bound states of two indirect excitons, obtained in the calculations by several groups [20–22], and a new phase in the multiexciton system [23]. The attractive exchange and van der Waals interactions may exceed the repulsive long-range dipole–dipole interactions between excitons if the distance between the quantum wells in a double well setup, and, correspondingly, dipole moments of indirect excitons, are not too large. Note, that from the analysis of the shape of the exciton emission band, the authors of the work [24] came to the conclusion that the interaction between indirect excitons is exclusively repulsive. In the papers [25,26] an alternative explanation of the experimental result obtained in Ref. [24] was given, which does not contradict the assumption of an attractive interaction. It was shown that the important contribution to the formation of the exciton emission bandshape at low densities comes from the traps, levels of which are localized under the exciton band. These traps become saturated at increased pumping and, thus, the emission, coming from the band proper, shifts to a higher frequency. 2. The finite value of the exciton lifetime plays an important role in the formation of the exciton condensed phase. Usually the exciton lifetime is much greater than the time of establishment of the local equilibrium in a system. But taking into account the finiteness of the exciton lifetime is necessary in the study of the spatial distribution of phases in two-phase systems, because the exciton lifetime is less than the time of the establishment of the equilibrium between different phases. The latter is determined by slow diffusion processes and is large. The finite exciton lifetime restricts the maximal size of the condensed phase domain and causes the existence of a correlation in positions of separate regions of the condensed phase. The suggested model [15–19] has succeeded in explaining the spatial structures observed in the luminescence of indirect excitons. Particularly, this model has allowed us to describe the spatial distribution of the excitons on the ring outside the laser spot, observed in Ref. [2], the spatial structure of the luminescence under the window in the metal electrode, observed in Refs. [3,4]. The involvement of Bose–Einstein condensation for excitons was not required for the explanation of the experiments. The Bose– Einstein nature of excitons contributes to the values of the phenomenological coefficients, which may be obtained in microscopic theory. But the condensation occurs in a real space due to
the interaction between excitons. The application of the presented model of the exciton condensation in the model of nucleation [15,17] is similar to a study of creation of electron–hole drops in semiconductors [27–29]. In this case, the islands of the exciton condensed phase are a two dimensional analog of the electron–hole drops. But, compared to the works [27–29], we took into account the correlation of the mutual positions of the islands, because they draw excitons from the same environment. This correlation allows us to explain the emergence of the periodicity in the distribution of the islands, which is observed in experiments. The possibility of using the system of indirect excitons in the optoelectronics is studied in several works [30–34]. The experimental possibility of building an exciton optoelectronic transistor [30], an excitonic integrated circuit [31] and an excitonic conveyor [32] was demonstrated recently. Therefore, the theoretical study of the propagation of the interacting system of indirect excitons in external fields is important. In this work we study the formation, amplification and passing through the barrier of the exciton condensed phase pulse in the semiconductor double quantum well in the setup, in which one of the electrodes has a slot, through which the excitons are created. We consider this problem using our model [15–19], approbated by the analysis of experimental results. The formation of the exciton condensed phase in the double quantum well under the slot excited by a steady uniform irradiation was studied in Ref. [19]. Unlike the work [19] this paper investigates the excitation of the exciton condensed phase by a pulse irradiation. This excitation creates an exciton condensed phase pulse, the movement and control of which are analyzed. The possibility of the amplification of a pulse by an additional laser pulse in the one-dimensional system was investigated in Ref. [35]. Compared to Ref. [35], in the presented paper, the processes of passing of the exciton pulse through a barrier and the possibility of stopping of the pulse by the barrier are investigated. Also we consider the real physical two-dimensional system (a slot in the electrode) and estimate the effect of exciton–exciton annihilation on the traveling pulse. The analysis of the difference in the movement of the pulses built from the excitons in a gas phase and the excitons in a condensed phase is presented.
2. Model of the system We consider the semiconductor structure with the double quantum well sandwiched between two metal electrodes. There is the slot in the upper electrode (Fig. 1). The width of the slot is 2b. Let us choose the OX axis along the slot, the axis OY in the transverse direction and the OZ axis along the normal to the electrode. Let the plane of the quantum well has a coordinate z ðz o 0Þ. The presence of the slot forms an inhomogeneous electric field that is additional to the homogeneous field of the solid electrode and gives an additional potential energy for excitons in quantum wells. The distribution of the exciton density in the double quantum well was determined [19] for the uniform steady-state excitation of the well through the slot. It was shown that a chain of periodically situated islands arises in the well below the center of the slot if the pumping exceeds some critical value. If the width of the slot is large, two chains of exciton condensed phase islands, located along the edges of the slot, appear. Additionally, the possibility of a movement of the chains along the slot was shown in Ref. [19] if driven by an external potential. In the considered system presented in Fig. 1, there are additional elements in comparison to the system investigated in the work [19]. We shall see, that these elements allow controlling the propagation of the exciton pulse.
Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i
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But in a certain vicinity of the borders of the slot the potential has a small minimum with a negative value. It appears due to the rearrangement of charges on the conductive electrode at the edges. The depth of the minimum increases if the quantum well is situated closer to the electrode with the slot.
3. Method of solution
Fig. 1. Setup of the system. The semiconductor double well structure is sandwiched between two metal electrodes. There is a slot in the upper electrode. U1 and U2 are the potentials on the upper and lower electrodes, correspondingly. Ug is the gate potential of the strip that creates a barrier for excitons. Arrows indicate the laser irradiation.
Fig. 2. Dependence of the potential created by the slot on the distance from the center of the slot in the plane of the quantum well for V 0 ¼ 5, z ¼ 15 and for different width of the slot: (1) b ¼ 10 (dashed curve), (2) b ¼7 (dotted curve), (3) b ¼4 (solid curve). The parameters are shown in dimensionless units, which will be introduced in the next section.
In the case of the strong electric field, created by electrodes, the potential energy Vtot for exciton in the quantum well, may be presented as V tot ¼ pz Eðy; zÞ, where pz is the exciton dipole moment, Eðy; zÞ is the electric field strength in the y; z that are point of the quantum well. This electric field is uniform in the region far from the slot Eðy; zÞ ¼ E0 . Additional potential for excitons, created by slot, is equal to V ðy; zÞ ¼ V tot V 0 , where V 0 ¼ pz E0 is the uniform shift of the exciton band by the electric field in the region distant from the slot. Using the solution of the problem for the electric field of a grounded conducting plane with a slot in an external electric field [36], and making the assumption that the distance between the electrodes is much greater than the width of the slot, we obtain the following equation for the slot contribution into the potential energy of an exciton: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V0 b z2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vðy; zÞ ¼ ð 1 þb =ξðy; zÞ 1Þ 2 2 2ξðy; zÞ2 1 þ b =ξðy; zÞ 0 13 2
y2 þ z2 þ b B C7 @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5; 2 2 2 2 2 2 ðy þ z b Þ þ4z b where
ð1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ξðy; zÞ ¼ 12 ½y2 þ z2 b2 þ ðy2 þz2 b2 Þ2 þ 4b2 z2 :
ð2Þ
The potential, created by the slot for exciton in the quantum well, is positive under the center of the slot (Fig. 2), because the electric field in this regions is less than the field far from the slot.
In this section we will concentrate our attention on the main equations for finding the excitonic density dependence on time and the position in space. In order to study the exciton distribution, let us consider the conservation law for the exciton density: ! ∂n n ! ¼ div j þ Gð r ; tÞ ; ∂t τ
ð3Þ
! where n is the exciton density, Gð r ; tÞ is the pumping (the number of excitons created in a unit time in a unit area of the quantum ! ! well), τ is the exciton lifetime, j ¼ M ∇ μ is the density of the exciton current, M is the exciton mobility. We use the Einstein formula M ¼ nD=kT for the exciton mobility, where D is a diffusion coefficient, k is the Boltzmann constant, T is the temperature. The described system is non-equilibrium due to the creation and decay of excitons. The system is also inhomogeneous. The assumption of the establishment of local equilibrium allows description of the system by a free energy depending on the exciton density, in its turn depending on the spatial coordinates. In such a way the free energy depends on the coordinates. Such an assumption is typical and important for the study of the non-equilibrium systems [37]. Having expressed the chemical potential as μ ¼ δF=δn, we chose the free energy in the Landau form: Z ! K ! 2 ð ∇ nÞ þ f ðnÞ þ nV tot : F½n ¼ dr ð4Þ 2 ! The term K=2ð ∇ nÞ2 characterizes the energy of inhomogeneity, f (n) is a free energy density of a uniform system, nVtot accounts for the potential energy in the external field. We approximate the density of the free energy by the following: n a b c ð5Þ f ðnÞ ¼ f 0 þ kTn ln 1 þ n2 þ n3 þ n4 ; n0 2 3 4 where the terms of the power series for the exciton density are important for large n and for the description of the condensed phase, while the term kTnðln n=n0 1Þ may be essential only if one wants to describe a system in which the density of excitons is small and the interaction between them is not important. Introducing dimensionless variables for length, density, energy and time: rffiffiffiffi rffiffiffi 2 K a d1 l u lu ¼ ; V u ¼ anu ; t u ¼ ; ð6Þ ; nu ¼ a c D and substituting Eq. (4) into Eq. (3), we obtain a nonlinear equation that governs the exciton density: ∂n ! ! ¼ d1 Δn þ ∇ ½n ∇ ð Δn þn ∂t n þb1 n2 þ n3 þV Þ þG gn2 ;
τ
ð7Þ
pffiffiffiffiffi where b1 ¼ b= ac o0, d1 ¼ kT=V u , and V is the potential created by the slot and given by Eq. (1). Let us do analysis of an influence of the external potential on the free energy. Since the phenomenological coefficients (a,b,c) in Eq. (5) describe the interaction between excitons, they should depend on the orientation of dipoles due to the dipole–dipole interaction. In order to create indirect excitons in double wells (to separate the charges) the strong electric field should be applied between electrodes. At low temperature, the strong external
Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i
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electric field orients exciton dipoles perpendicular to the wells' plane independently on the phase state of excitons and the screening does not occur (dipoles are already oriented entirely). The potential V, created but the slot, is much smaller than the potential of uniform electric field far from the slot V0 ðjVj⪡jV 0 jÞ. So, it does not affect the dipole orientation, and, therefore, the coefficients of the free energy (a,b,c), which are the same anywhere in the quantum well. The exciton–exciton annihilation (Auger process) was taken into account by introducing the term gn2 in Eq. (7). Eq. (7) is a non-linear phenomenological equation that describes the distribution of exciton density taking into account ! the pumping Gð r ; tÞ and the finite lifetime τ.
Fig. 3. Time evolution of the exciton density maximum for different sets of parameters. Squares correspond to the exciton pulse in the condensed phase ðP ¼ 0:64; g ¼ 0Þ, triangles correspond to the exciton pulse in the gas phase ðP ¼ 0:1; g ¼ 0Þ, circles correspond to the exciton pulse in the condensed phase taking into account Auger recombination of excitons ðP ¼ 0:64; g ¼ 7 10 4 Þ, stars correspond to the exciton pulse in the gas phase taking into account Auger recombination of the excitons ðP ¼ 0:1; g ¼ 7 10 4 Þ. b ¼ 4, β ¼ 10, Δt ¼ 20, τ ¼ 1000, b1 ¼ 2:23, d1 ¼ 0.2.
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4. Creation and movement of the exciton condensed phase pulses A domain in which the exciton density is high can be formed by illuminating the system with a laser pulse. We assume the excitons are generated during a time period Δt with the position dependent rate: ! 8 2 2 > < P exp ðx x0 Þ þ ðy y0 Þ if t t 1 o Δt; jy y0 j o b; Gðx; y; tÞ ¼ β2 > : 0 in all other cases: ð8Þ Eq. (8) takes into account the non-transparency of the electrodes. P is proportional to the number of excitons created by the laser per unit time in a unit area of the quantum well plane. The laser pulse causes heating. Considering this problem, we assume that the time by which the temperature reaches equilibrium is much less than the exciton lifetime τ. The exciton density in the domain may be sufficient for the formation of the condensed phase. We will refer to such domains are an exciton condensed phase pulse. In the opposite case of the exciton density insufficient for the condensation the domain will be referred to as the exciton gas pulse. Time evolution of two types of pulses is different. This difference can be illustrated by considering the evolution of the exciton density with time for standing pulses, id est when there is no force driving excitons to move along the slot. First, we consider the decay of the pulse neglecting the exciton–exciton annihilation (g ¼0). The dependence of the peak values of exciton density in the pulses is presented in Fig. 3 for two magnitudes of the laser pulses: P¼ 0.1, g¼ 0 and P ¼ 0:64; g ¼ 0. If the value of the amplitude of the laser pumping (P ¼0.1) is less than the threshold value, the exciton pulse is formed in the gas phase. It is seen from Fig. 3 that in this case the peak of the exciton density in the pulse decreases rapidly with time after the generation stops, and the width of the exciton pulse increases. If the value of the amplitude of the generation pulse exceeds the threshold value (P¼0.64), the excitons in the pulse form the condensed phase. The peak value of the exciton density in
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Fig. 4. Spatial distribution of the exciton density at b¼ 4, P¼ 0.64, β ¼ 10, τ ¼ 1000, b1 ¼ 2:23, d1 ¼ 0.2, δ ¼ 0:09 at different moments of time after switching off the pumping: (a) t ¼ 100, (b) t¼ 500, (c) t ¼900, (d) t ¼1300. The white lines show the edges of the slot.
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this case remains approximately constant during the exciton lifetime. Therefore, the attractive interaction between excitons in the condensed phase inhibits the spread-out of the domain. As the excitons decay with time, the width of the pulse decreases, it thins down, but the density maximum preserves the same value over long time till the system is found in the condensed phase state. To study the effect of the annihilation, we have chosen a large value of the annihilation rate, such that the lifetimes due to the annihilation and radiation are the same. It is seen from Fig. 3 that annihilation affects two types of pulses in the qualitatively different ways. For the condensed phase pulse ðP ¼ 0:64; g ¼ 0:0007Þ, the duration of the period, in which the maximum of the exciton density remains constant, decreases compared to the case when annihilation is neglected ðP ¼ 0:64; g ¼ 0Þ. The gas phase pulse is affected by the annihilation weakly as seen from comparison of the time dependence of the density profiles for P ¼ 0:1; g ¼ 0:0007 and P ¼ 0:1; g ¼ 0. Such behavior is obviously caused by the fact that the exciton density in the condensed phase is larger. The next problem we have considered is the traveling of the pulse. We assume that the excitons are driven along the slot by a bias represented in the free energy by an additional potential V l ðxÞ ¼ δx;
ð9Þ
where δ is a constant. Such potential can be created, for example, by using an electric field, as it the authors [38] did, or by applying an inhomogeneous stress to the crystal. Four snapshots of the exciton pulse taken at different moments of time are shown in Fig. 4. Additionally, Fig. 5 plots the projections of density distribution onto the XZ plane. One can see that the exciton condensed phase pulse moves along the slot thinning down with time as the excitons decay. The velocity of the propagation depends on the potential ðv δD=kTÞ. Provided the exciton pulse is in the condensed phase, the maximum value of the exciton density in the pulse remains almost constant during its lifetime, while the width of the pulse decreases. In the case of the gas phase pulse, the profile spreads out. Two exciton pulses are formed in the case of a wide slot. They also move along the slot parallel to each other. Exciton pulses can be used for transmitting information. The condensed phase pulses may transfer signal to greater distances, if the detector at the output reacts to the amplitude of the exciton density.
Fig. 5. Dependence of the exciton density on the x coordinate at different moments of time in the system without the barrier (parameters are the same as in Fig. 4).
5
5. Amplification of the exciton condensed phase pulse by a laser pulse An exciton condensed phase pulse can be amplified by an additional laser pulse. We have shown already that the exciton density within the condensed phase remains constant during the exciton lifetime, but at a certain moment of time the amplitude of the pulse begins to drop down and the system of excitons in the pulse transforms to the gas phase. To recover the exciton condensed phase pulse, let us shine an additional laser pulse at a certain fixed moment of time. The amplitude of this additional pulse may be lower than the threshold value for the creation of the condensed phase pulse. Matching the position of the additional laser pulse with the position of the traveling exciton pulse leads to the transition of the exciton pulse to the condensed phase. Thus, two laser pulses in a sequence help to maintain the condensed phase, the exciton pulse exists for a longer period of time and can travel further. The time dependence of the maximum values of the exciton density in the first, the second and resulting pulses is plotted in Fig. 6. If the detector reacts to the amplitude of the exciton pulse, then the signal carried by excitons in the condensed phase can be detected at greater distances.
6. Control of the exciton condensed phase pulse using the barrier The propagation of the exciton condensed phase pulse can be controlled, for example, using a barrier. The barrier can be created by placing a thin metal strip perpendicularly to the slot (see Fig. 1). The barriers of different magnitudes can be formed by applying voltage to the strip. Let us approximate the additional potential for excitons created by the strip as follows: V b ðxÞ ¼ S expð ðx x0 Þ2 =2ðΔxÞ2 Þ;
ð10Þ
where S is the magnitude of the barrier and Δx is the width of the barrier. The cumulative potential for excitons created by the linear potential Eq. (9) and the barrier Eq. (10) is shown in Fig. 7, the parameters of these potentials are δ ¼ 0:09, S¼ 1. One can observe that in some cases (S ¼1) an obstacle is created on the way of the traveling exciton condensed phase pulse. Let us analyze the effect of the height of the barrier on the penetration of the pulse through the barrier. The comparison of the projections of the exciton density onto the YZ plane for S ¼0
Fig. 6. Dependence of the exciton density maximum (nmax) on time (t). Squares correspond to the first exciton pulse (P¼ 0.64), triangles correspond to the second pulse (P¼ 0.1), circles correspond to the total pulse and the dashed line corresponds to the maximum of the decreasing exciton density without additional laser pulse.
Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i
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Fig. 7. Total additional potential for excitons created by linear potential and barrier ðV l ðxÞ þ V b ðxÞÞ, δ ¼ 0:09, S ¼1.
Fig. 8. Dependence of the exciton density on the x coordinate at different moments of time in the presence of the barrier at S ¼0.2, Δx ¼ 2:24, δ ¼ 0:09, b ¼ 4, P ¼0.64, β ¼ 10, Δt ¼ 20, τ ¼ 1000, b1 ¼ 2:23, d1 ¼0.2.
Fig. 9. Dependence of the exciton density on the x coordinate at different moments of time in the presence of the barrier, S ¼0.3, Δx ¼ 2:24, δ ¼ 0:09 (other parameters are the same as in Fig. 8).
(Fig. 5) and S¼ 0.2 (Fig. 8) shows that the amplitude of the pulse after passing the barrier drops down with increasing the height of the barrier. For large S the exciton condensed phase pulse does not survive passing the barrier (Fig. 9), the pulse in the condensed phase exists beyond the barrier only at S r0:3. The dependence of
the exciton density on spatial coordinates in the case of the barrier with parameters S ¼1, Δx ¼ 2:24 and the position of the maximum x0 ¼ 100 is presented as the two-dimensional plot in Fig. 10 and as the projection onto the YZ plane in Fig. 11. The position of the pulse does not change with time after the pulse reaches the barrier. The exciton condensed phase pulse stops in the area before the barrier. This demonstrates the possibility of controlling the motion of the pulse by changing the barrier parameters. As the pulse stops, it changes the shape (see Fig. 10). The collective penetration of excitons through a barrier is somewhat similar to the tunneling of a quantum-mechanical particle through a barrier: the probability of the tunneling decreases for higher barriers as well. The dependence on the width of the barrier (on Δx) is not straightforward, since we considered the potential for excitons as consisting of two parts: a linear part of the driving force and the potential of the barrier. Depending on the values of Δx and δ the total height of the barrier may either decrease or increase as a function of Δx (see Fig. 7) with different effects on the pulse penetration. We have carried out our calculations in dimensionless units. Let us perform some estimates to represent the results in real dimensional quantities. We chose the following values of parameters: τ ¼ 100 ns, T ¼3.7 K, nu ¼ 2 1010 cm 2 , D¼9 cm2/s, anu ¼1.6 meV. In this case the width of the slot 2b is 5:36 μm (b¼4). The traveling path of the exciton condensed phase pulse during its lifetime is l ¼ 67 μm, and the height of the barrier stopping the pulse is S ¼ anu ¼ 1:6 meV (S ¼1). The narrowing of the pulse with time is indicative of the attractive interaction between the particles forming the pulse. This process differs from the Bose–Einstein condensation, in which the narrowing of the distribution occurs in the momentum space, that should lead to a broadening of the spatial width. Therefore, an experimental study of the behavior of the domain may determine the nature of the condensed phase.
7. Exciton autosolitons So far we have studied the exciton density pulses localized in space. These pulses are created by laser pulses acting during a certain time period. Exciton density pulses created this way gradually weaken and finally disappear. However, the localized in space exciton pulses (autosolitons) are possible at stationary excitation by irradiation in a certain interval of the exciton generation rate as it was shown for the one-dimensional system [25,26]. Let us consider the possibility of the formation of the exciton autosolitons in the potential created by the electrode with the slot. As shown in Ref. [19], the periodical distribution of the exciton density in the quantum well emerges for the stationary irradiation below the slot in the electrode in a certain interval of the pumping Gc1 o G oGc2 . The homogeneous distribution of the exciton density exists outside this interval of the exciton generation rate. We showed that in this case of two-dimensional system at steady-state pumping there is a spatially nonuniform solution of Eq. (7) for G oGc1 in the form of an isolated peak as in one dimensional system [25,26]. This solution may be obtained by solving Eq. (7) with the generation rate consisting of a constant value G0 and an additional pulse dG with a maximum at a certain point in space and at the time moment dG ¼ s exp½ wððx x0 Þ2 þ ðy y0 Þ2 Þ exp½ pðt t 0 Þ2 ;
ð11Þ
where s; w; p are parameters. Eq. (11) describes a pulse of the pumping acting during a period of time and having the maximum at the point x0 ; y0 .
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Fig. 10. Spatial distribution of the exciton density for S ¼1, Δx ¼ 2:24, δ ¼ 0:09 (other parameters are the same as in Fig. 8) at different moments of time after switching off the pumping: (a) t¼ 100, (b) t¼ 500, (c) t ¼900, (d) t ¼1300. The white lines show the edges of the slot.
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Fig. 12. Bright autosoliton. Spatial distribution of the exciton density for G ¼ 0:005 o Gc1 , b ¼7, b1 ¼ 2:23, x0 ¼ 60, y0 ¼ 30. Fig. 11. Dependence of the exciton density on the x coordinate at different moments of time in the presence of the barrier, S¼ 1, Δx ¼ 2:24, δ ¼ 0:09 (other parameters are the same as in Fig. 8).
The solution of Eq. (7) for the system of indirect excitons under the slot in the metal electrode obtained in the case of imposing the addition pulse Eq. (11) is presented in Fig. 12. The solution exists at t-1, i.e. at the times, when the action of the addition pulse is already absent. The shape of the peak nðx; yÞ does not depend on parameters s; w; p in some region of their values, but this solution exists in a finite interval of the pumping ðG ¼ 0:0048 0:0052Þ. In addition, the solution in the form, presented in Fig. 12, arises also, if the additional pulse is absent, but there is distribution of the exciton density in the initial moment of time in the form: nðx; y; 0Þ ¼ s expð wððx x0 Þ2 þ ðy y0 Þ2 ÞÞ:
ð12Þ
Such localized spatial distribution of the exciton density is called “bright autosoliton”. According to the classification of Ref. [39] it belongs to the class of static soliton. Localized solutions exist also in a certain region of pumping greater the value, at which the periodical structure arises ðG 4 Gc2 Þ. The dependence of the exciton density may be obtained from Eq. (7) choosing an additional pumping pulse in the form Eq. (11),
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Fig. 13. Dark autosoliton. Spatial distribution of the exciton density for G ¼ 0:008 4 Gc2 , b ¼7, b1 ¼ 2:23, x0 ¼ 60, y0 ¼ 30.
but with s o 0. Such solution is presented in Fig. 13. These structures can be called “dark autosolitons”. They exist in a certain region of pumping ðG ¼ 0:0078 0:0081Þ. If there is an external field in the system creating a linear potential (see Eq. (9)), the autosoliton moves along the slot. So, these autosolitons can be
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used for transmitting information as well. In this case, the motion occurs at steady-state irradiation, but such pulse does not decay with time. 8. Conclusions We have considered the dynamics of the exciton condensed phase pulse, created in double quantum wells by the external irradiation. The modeled system consists of a double well structure sandwiched between two metal electrodes, the top one having a slot that creates an additional potential for excitons. The spinodal decomposition approach, generalized to the case of particles with a finite lifetime, was used to describe the evolution of the exciton density distribution in condensed and gas phases. To ensure traveling of the exciton pulse, an additional potential linearly depending on the coordinate along the slot was included. The possibility to control the propagation of the exciton pulse was demonstrated using either the barrier, which arises due to the potential created by a metal strip, or the additional laser pulse. The numerical simulations produced the following results: 1. Exciton condensed phase pulse arise if the intensity of the laser pulse is greater than the threshold value dependent on the parameters of the system. 2. In the case of a narrow slot, an exciton condensed phase pulse is formed in the center of the slot, and in the case of a wide slot, the exciton condensed phase pulse splits into two pulses, located near the boundaries of the slot. 3. In the presence of the linear driving potential along the slot, the exciton pulse moves with a constant velocity. Maximum density remains constant during the exciton lifetime and the pulse does not spread out if the exciton system is in the condensed phase. The width of the pulse decreases with the time. 4. The attenuated pulse can be amplified by imposing an additional laser pulse thereby increasing the path traversed before the disappearance. 5. The exciton pulse can be stopped or its shape can be changed by creating a potential barrier for excitons. With increasing the height of the barrier, the permeation of the excitons through the barrier decreases and the shape of the pulse changes stronger. 6. The existence of the bright and dark autosolitons in the system of excitons in double quantum wells under the slot in the electrode at steady pumping was demonstrated. The paper analyses how the exciton motion in inhomogeneous external fields is influenced by the collective effects leading to the formation of the condensed phase. The experimental observation of the pulse behavior will allow us to determine the origin of the condensed phase. If the condensed phase is caused by attractive interaction between excitons, the spatial width of the pulse should decrease during the existence of the condensed phase. In the case of the Bose–Einstein condensation the narrowing of the pulse should occur in momentum space, that leads to a broadening of
the spatial width. Also the spatial increase of the width of the pulse must occur for the pulse of excitons in the gas phase. Moreover, the investigations of the exciton pulses may be perspective for application of the exciton systems in the optoelectronics.
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Please cite this article as: O.I. Dmytruk, V.I. Sugakov, Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.11.055i