Superlattices and Microstructures 33 (2003) 103–115 www.elsevier.com/locate/jnlabr/yspmi
Electric field effects on the magnetoexcitons in quantum wells Ecaterina C. Niculescu Department of Physics, “Politehnica” University of Bucharest, 313 Splaiul Independentei, RO77206 Bucharest, Romania Received 27 September 2002; received in revised form 26 June 2003; accepted 2 July 2003
Abstract In the framework of the effectivemass envelopefunction theory, the electronic properties of a rectangular quantum well (QW) in the presence of a strong magnetic field in the QW plane and a crossed electric field parallel to the growth direction are studied. The single particle states for electrons and holes and the excitonic binding energy are obtained for several magneticfield values, as functions of the electric field strength. The electric field breaks down the degeneracy of the states symmetrically positioned in p space, leading to a nonparabolic subband structure. We found that for the exciton moving in the QW plane, for some applied field strengths, (i) the binding energy exhibits a maximum and (ii) a transition of the ground state from a zero inplane centre of mass momentum to a finite momentum occurs. © 2003 Elsevier Ltd. All rights reserved.
1. Introduction Stimulated by the rapid progress in nanometerscale fabrication technology, a large number of papers focused on the optical properties of lowdimensional systems have been published. These structures may create new physical phenomena and they show a great potential for device applications. The first step towards understanding the optoelectronic properties of a quantumwell (QW) device is to investigate the quantumconfined electron and hole states in the well. In these studies one can obtain further insight by making use of perturbations such as external electric or magnetic fields or the biaxial strain in latticemismatched QWs. Variational [1] and exact calculations using both analytic Airy function methods [2–4] and numerical methods [5–7] for the Stark shifted energy levels and the resonance width of quasibond states in a single QW have been reported. Some authors
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[8–11] discussed the influence of the external electric field applied in the growth direction on the excitonic transitions between different conduction and valence subbands. Asymmetric Stark shifts were found in Inx Ga1−x As/GaAs nearsurface QWs, which are due to the image charge effect and the Coulomb interaction [12]. Also the effect on exciton binding energy when the exciton is electrically tuned from a 2D exciton to a spatially separated electron–hole pair have been investigated [13]. Magnetospectroscopy has proven to be a powerful method for the study of a variety of excitations in lowdimensional semiconductor structures, and experimental [14, 15] and theoretical [16–18] works of the electronic structure of QWs under a magnetic field have been published. The excitonic levels, binding energy, and optical transition probabilities in symmetric [19] and asymmetric [20] double QWs as a function of barrier and well widths and the magnetic field strength were studied. Recently, Riva and Peeters [21], using a variational method, calculated the spinsinglet and spintriplet states of a negatively charged exciton confined to a single QW in the presence of an external magnetic field. Maialle and Degani [22] have investigated the electron–hole exchange interaction for free excitons in QWs with an applied transverse magnetic field. They found that the magnetic field introduces an anisotropy on the exchange interaction with respect to the relative directions of the field and the exciton centreofmass momentum. The kinetics of the excitons in GaAs/Alx Ga1−x As coupled QWs in the presence of an external inplane magnetic field has been studied experimentally [23] and theoretically [24]. In this system, a shift of the exciton ground state from a zero inplane centre of mass (CM) momentum to a finite inplane momentum was reported. Experimental and theoretical works on the effects of both electric and magnetic fields in lowdimensional structures have also been the subject of interest in recent years. Optical spectroscopy of QWs with F and B applied along the growth direction was studied by Zawadzki [25]. Also in this geometry, Dzyubenko and Yablonskii [26], using a nonvariational method, studied the excitonic levels for symmetric and almost symmetric double QWs as a function of the electric and magnetic field strengths. Citrin and Hughes [27] have calculated the Franz–Keldysh effect on Landau levels and magnetoexcitons in the presence of a strong dc electric field in the QW plane and a crossed static magnetic field B parallel to the axis of growth of the QW structure. In [28] we developed a method to calculate the electronic structure of cylindrical semiconductor pills subjected to an axial magnetic field and an electric field with different orientations. We calculated e1–hh1 transition energies by solving for the exact magnetoelectronic subband wavefunctions and we treated the electric term by a variational approach. In this paper we present an analytical method to calculate the exact single particle states for electrons and holes confined to a QW with F parallel to the growth direction and a strong magnetic field in the QW plane. In this geometry, we find the electron–hole pair binding in the ground state using a variational approach. For simplicity, we assume the infinite potential well and use the simpleparabolicband model for electron and hole, neglecting the mixing effect of heavy and light hole states.
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2. Theory 2.1. Single particle states We consider a GaAs QW of width L, with a constant magnetic field in the QW plane, B = B xˆ , and a crossed electric field aligned with the growth direction, F = F zˆ . In the single band effectivemass approximation, the electron Hamiltonian is given by He =
(pe + eA)2 + Ve (z) + eFz 2m ∗e
(1)
where e = e and A are respectively the elementary charge and the magnetic potential vector, and Ve (z) is the infinite confinement potential in the growth direction. Choosing the Landau gauge A = (0, −Bz, 0), the Hamiltonian becomes He =
p2e (eB)2 2 eB + z − ∗ zp ye + Ve (z) + eFz. ∗ 2m e 2m ∗e me
(2)
For this single particle Hamiltonian, motion along the magnetic field direction (the x direction) is separable from the motion in the (y, z) plane. We find the electron eigenstates with energies E ne by solving ∂2 (eB)2 2 eB 2 ∂2 e e H2D ψn = − ∗ + 2 + z − ∗ zp ye + Ve (z) + eFz ψne 2m e ∂y 2 2m ∗e me ∂z = E ne ψne . e , p ] = 0, the solution for the wavefunction is Since [H2D ye i e p ye y f ne (z). ψn (y, z) = exp
(3)
(4)
By substituting (4) into Schr¨odinger Eq. (3) we obtain the equation for the envelope wavefunction of the electron with momentum component p ye : 2 2 d2 f ne (z) m ∗e (ωce )2 F e 2 e ∗ F (z − z 0 ) f n (z) + Ve (z) + p ye − m e 2 fne (z) + − ∗ 2m e dz 2 2 B 2B = E e f ne (z).
(5)
Here ωce = eB/m ∗e is the cyclotron frequency and z 0e , the centre of the cyclotron orbit, is given by m ∗e F . (6) eB 2 In the absence of the quantum confinement, the solutions to Eq. (5) are those of a displaced harmonic oscillator with 1 F F2 (7) E n2D = n − ωce + E S ( p ey , B, F) = E nL + p ye − m ∗e 2 2 B 2B z 0e = p ye −
where E nL are the Landau levels and n = 1, 2, 3 . . ..
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In the QW case, the subband energies are obtained by means of the boundary conditions L e e L fn − = fn = 0. (8) 2 2 is the Assuming that f ne (ξ ) = exp(−ξ 2 )Φne (ξ ), where ξ = (z − z 0e )/lc and lc = eB magnetic length, Eq. (5) becomes dΦne (ξ ) d2 Φne (ξ ) + λen Φne (ξ ) = 0 − 2ξ dξ dξ 2 with λen =
2 ωce
E ne − p ye
F F + m ∗e 2 B 2B
(9)
− 1.
(10)
If F = 0 and p ye = 0, the solutions for this equation are the confluent hypergeometric functions e λn 1 2 e Φn (ξ ) = H y − , ; ξ (for the even states) 4 2 or
Φne (ξ )
= Hy
1 λen 3 2 − , ;ξ 2 4 4
(for the odd states).
The eigenenergies are determined by solving Eq. (8). If F = 0 and (or) p ye = 0, the solutions Φne (ξ ) are given by e λ 1 1 λen 3 2 Φne (ξ ) = C1 H y − n , ; ξ 2 + C2 H y − , ;ξ . 4 2 2 4 4
(11)
The energies of the single particle states and the energy dispersion relations in different magnetoelectronic subbands are obtained by substituting this expression into f ne (ξ ) and changing the energy E ne to obtain the zeros in Eq. (8). In the simplest approximation of parabolic bands, neglecting the mixing between heavy and light holes, the eigenstates of the heavy hole in a QW can be calculated by a similar procedure. 2.2. Exciton states Excitons can be described by the following twoparticle Hamiltonian ex e h H3D = H2D ( p ye ) + H2D ( p yh ) −
where r = re − rh = (ρ, z) and and reduced mass, respectively.
1 µ
=
2 ∂ 2 e2 − 2 2µ ∂ x εr
1 m ∗e
+
1 m ∗h
(12)
are the electron–hole relativecoordinates
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The Hamiltonian Eq. (12) can be rewritten using the coordinates for the CM motion (R, P) and relative motion (RM) (ρ, p) R = δe ρ e + δh ρ h ; with δe = obtain ex H3D
m ∗e /M, δh
=
P = pe + ph ; m ∗h /M
and M =
P2 p2 eB − z PY − eB = Y + 2M 2µ M +
p = δh pe − δe ph
m ∗e
+
m ∗h
(13)
is the total mass of the exciton. We
ze zh + ∗ m ∗e mh
e2 B 2 py + 2
2 2 pzh pze e2 . + + V (z ) + V (z ) + eFz − e e h h 2m ∗e 2m ∗h εr
z h2 z e2 + m ∗e m ∗h
(14)
For the groundstate exciton, we use a separable trial function [8] Ψ ex ( e ,
h)
= ψ1e (z e )ψ1h (z h )exp[−ρ/α]
(15)
i where ψ1i (z i ), i = e, h are the lowest eigenfunctions of H2D , and α—the spatial extension of the exciton wavefunction in the QW plane, is variational parameter. The assumption of a separable trial function introduces some limitations in the validity of the results. This wavefunction is not valid [8] for QW wides larger than the 3D ˚ However, it must be a good approximation for QW in exciton diameter (d0 ∼ = 300 A). high magnetic field, when the wavefunctions compression provided by the field becomes important, and also if the applied electric field is not excesively large, such as the radius of the exciton orbit
α(F, B) < d0 /2.
(16)
In the following we will consider three electric field regimes, weak, intermediate and strong, for F values such that Eq. (16) holds. Finally, the binding energy of the exciton is obtained as ex E b = E 1e + E 1h − minΨ ex H3D Ψ ex . λ
(17)
3. Results and discussion To model GaAs QW structure, we use ε = 13.1, m ∗e = 0.067m 0 and an isotropic heavy hole mass, m ∗h = 0.35m 0. This isotropic model should be appropriate for large wells at high magnetic fields [29], when the quantum confinement is weaker and the splitting of hole states is less important than the magnetic squeezing. 3.1. Single particle states First, we investigated the effects of the applied fields on the electron and heavy˚ GaAs QW. In Fig. 1 we present the normalized hole subband states for an L = 300 A e = E e /E (o) vs the electric field for p ye = 0 and various electron ground state energy E 1 1 B. Here E (o) = 2 /2m ∗e (π/L)2 is the ground state energy of an infinite QW at zero applied fields. We have studied the high magnetic field regime, when the well width is
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3
Normalized energy
pye = 0
B = 20 T
2 1 0 1
B = 10 T
B = 15 T
2 3 0
20
40
60
80
F (kV / cm) e = E e /E (o) of an electron in a GaAs QW of L = 300 A ˚ wide vs the Fig. 1. Normalized ground state energy E 1 1 applied electric field. The arrows indicate the crossover from low to high electric field regime. For comparison, the energy of a free, 3D electron, is also shown (dashed curves).
larger than the magnetic length, and the barrier constraint is a small perturbation on the magnetic energy. In order to understand the electronic properties in detail, we analysed the particularly interesting regime of electric fields such that the Stark shift E S  (Eq. (7)) varies from smaller than the ground magnetoelectronic energy E 1e (F = 0) to larger than this quantity. The exact electron energies are compared with the corresponding 3Dlimits (Eq. (7)). Notice that at F = 0 in the weak quantum confinement regime, the ground state energy approaches the n = 1 Landau level. As Fig. 1 shows, this highmagnetic field limit is effectively reached at B = 20 T. In the Ffield presence, the confinement is a complicated competition between wavefunction compression provided by the magnetic field, the electric field induced polarization and the potential barrier effects. For convenience, in Fig. 1, the normalized energy axis (y axis) is roughly divided into three intervals: from e to 1, from 1 to −1, and from −1 to −3, which are referred to as the low electric field E 1max regime, the intermediate, and the high electric field regime, respectively. For each curve, the arrow indicates the value of Ffield where the ratio η = E/E 0L = (m ∗2 /e)(F 2 /B 3 ), which describes the competition between the magnetic field and the electric one, is equal to unity. This is the crossover regime where F and B are comparable. Beyond this critical value (which is B dependent), the electric field effect predominates and the barrier constraint appears. Notice that the energy separation between the exact and the 3D energies (dashed curves) increases with F. This is associated with electric field induced deformation of the electron wavefunction in the QW. The effect of the potential barrier is weak in the magnetic confinement regime when wavefunction is compressed inside the QW. In contrast, for large F, f e (z) shift near the left well edge, and the energy depends more strongly on the barrier constraint.
E.C. Niculescu / Superlattices and Microstructures 33 (2003) 103–115
(a)
109
100
pye = 0
Energy (meV)
c
c
50 b
b
a
0
a
50 0
20
40
60
80
(b) c
pyh = 0
15
Energy (meV)
c
10
b
b
5
a a
0
5
0
5
10
15
20
25
30
F (kV / cm) ˚ wide GaAs QW, at B= 10 T (dashed lines) Fig. 2. The single particle electron (a) and hole (b) energies in a 300 A and B = 20 T (solid lines). The E 1 , E 2 and E 3 energy levels indicated by a, b, and c on the lines, respectively are given as functions of the electric field.
Fig. 2 shows the Stark shift of the lowest magnetoelectronic subbands for p ye(h) = 0. The two particles are differently influenced by an electric field. For small and intermediate values, the electron is only slightly affected by the electric applied field and a parabolic behaviour of the electron energy with F is observed (Fig. 2(a)). As the hole is a rather heavy particle, the effect of the electric field is more pronounced, leading to an almost linear behaviour of the Stark shift (Fig. 2(b)). As a result, the crossover regime occurs at smaller Ffield values than those in Fig. 1 (Fch ∼ = 4 kV cm−1 for B = 10 T and −1 Fch ∼ = 11 kV cm for B = 20 T). The polarization effect is weaker for B = 20 T,
110
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 pyh  = 0.1 ( h/nm)
B = 10 T
15
c
Energy (meV)
10
c b
5 a b
0 a
5
0
5
10
15
F (kV / cm) Fig. 3. The energy levels for a hole moving perpendicularly to the magnetic field in a GaAs QW at B = 10 T. The hh1, hh2 and hh3 states indicated by a, b, and c on the lines, respectively are given as functions of the electric field for p yh = 0.1(/nm) (solid lines) and p yh = −0.1(/nm) (dashed lines).
when the electron and hole densities are more squeezed perpendicular to the magnetic field. The behaviour of the subband energies for a hole moving perpendicularly to the magnetic field, shown in Fig. 3, is more complicated. There are some observable trends. (i) At zero electric field, the subband energy is degenerate for the carrier states corresponding to symmetrical values of p yh . The electric field breaks down the degeneracy leading to a strong anisotropy of the subband structure. (ii) The energy levels of the hole with p yh > 0 first increase with F then decrease. It is a result of the competition between the Lorentz force (term in the single particle Hamiltonian proportional to Bz), which pushes the hole to the left side of the QW, and the electric field effect, which induces a spatial shift to the right direction. Thus, in the low electric field regime up to the crossover regime, the hole wavefunction becomes more concentrated at the centre (see Fig. 4(a)). The linear term in the energy shift E S ( p yh , B, F) increases with F and determinates the enhancement of the subband energy. In the strong electric field regime, the contribution of the quadratic term in E S becomes dominant. An increase of this negative term leads to a reduction in the subband energy. (iii) For the states with p yh < 0, the energy levels strongly decrease with F because both the Lorentz force and the electric field produce the shifting of the wavefunction amplitude toward the right edge of the well (see Fig. 4(b)). The linear and the quadratic contributions previously mentioned are negative and the subband energy decreases monotonically with F (Fig. 3). In addition, from Fig. 3 it can be seen that at high electric fields the crossings between some energy levels are possible. This is a result of the coupling and competition of the barrier potential with the electric one. The effect of the two potentials on energy levels is related to the corresponding wavefunctions. At F = 0, the hh1 and hh3 wavefunctions
E.C. Niculescu / Superlattices and Microstructures 33 (2003) 103–115
111
(a)
1 0.5
fh
10 7.5
0 5
10
F (kV / cm)
0
z (nm)
2.5 10
0
(b)
1
fh
8
0.5 6
0
F (kV / cm)
4
10 0
z (nm)
2 10
0
Fig. 4. The ground state wavefunction of a moving hole with (a) p yh = 0.1(/nm) and (b) p yh = −0.1(/nm) in a GaAs QW subjected to a magnetic field of 10 T.
have their maximum near the well centre, while the hh2 state has the highest probability density approximately midway between the centre and the well edge. In the presence of the applied electric field, when the hole moves towards the right well edge, the hh2 wavefunction is strongly distorted by the barrier potential. Thus, the hh2 subband energy changes more significantly with increasing F than the others.
E.C. Niculescu / Superlattices and Microstructures 33 (2003) 103–115
10
Binding energy (meV)
(a)
12
Eex (meV)
112
8
6
8
4
20 T 0 0
5
10
15
20
F
4
15 T
B = 10 T
2 0
5
10
15
20
25
30
F (kV/cm)
(b)
7
20

 (nm)
Binding energy (meV)
5
d
a
15
6
c b
10 5
PY = – 0.1 0
4 a b
0
5
10
P Y = 0.1
15
20
25
30
F
3 c
2
d
B = 10 T
0
5
10
15
20
25
30
F (kV/cm) ˚ GaAs QW, B = 10 T vs electric field. (a) PY = 0; p y = 0. Fig. 5. Binding energy of the exciton in an L = 300 A In the inset, the exciton energy is plotted. The solid dots are the experimental results of [30], for F = 0. (b) Exciton with PY = −0.1(/nm) and a: p y = −0.1δh (/nm); b: p y = 0.1δe (/nm); exciton with PY = 0.1(/nm) and c: p y = 0.1δh (/nm); d: p y = −0.1δe (/nm). Inset: The mean electron–hole distance along the z direction. The same curve conventions are used as in the main figure.
3.2. Exciton states ˚ GaAs QW as a In Fig. 5 we plot the binding energy of the exciton in an L = 300 A function of the electric field. For PY = 0; p y = 0 (Fig. 5(a)) we observe the wellknown behaviour [8] that as F increases the binding energy decreases. This is a consequence of the electric field induced spatial separation of the wavefunctions corresponding to
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the e1 and hh1 states. The effect is less pronounced for B = 20 T, due to an additional magnetic confinement. For such a system, the Stark shift of the exciton energy appears to be quadratic with applied electric field (inset of Fig. 5(a)). For F = 0, our numerical results agree reasonably with the experimental [30] and theoretical [21] values (see the circles in the inset of Fig. 5(a)). Fig. 5(b) shows the results for the exciton moving in the QW plane perpendicularly to the magnetic field. Note that for a magnetic field range from 8 to 12 T, in the absence of an electric field, Maialle and Degani [22] obtained E b ∼ = 6 meV for an ˚ exciton moving perpendicularly to the magnetic field with PY = 0.1 nm−1 in a 250 A wide GaAs/Al0.3 Ga0.7 As QW. Our result for F = 0 slightly overestimates this value, which is probably a consequence of the infinitewell approximation. From this figure, we find that the exciton binding energy is asymmetric with respect to the CM momentum component PY . For PY < 0, the binding energy quickly decreases with F. For PY > 0, E b increases to a maximum, and then decreases continuously by increasing the electric field. This can be understood as follows: in a first approximation, the binding energy comes from the Coulomb interaction between the electron and the hole (the last term in Eq. (12)). For large magnetic fields, as both electron and hole are spatially confined into a ringlike area with cyclotron radius lc L/2, this term has roughly a 1/z e − z h  ∼ = 1/z e0 − z h0  dependence with increasing electric field. In the CM coordinates, using Eq. (6), we obtain PY M F ∼ − z eh  = . (18) eB eB 2 We can see that for PY < 0, both the Lorentz force and the electric field spatially separate the electron and hole, which causes a diminution of the exciton binding energy. If PY > 0, for small F the electron–hole distance along the z direction decreases, which leads to an enhancement of the binding energy. This behaviour continues until the electric field reaches a value around Fmax = B PY /M, where z eh  ∼ = 0 and there exists a peak of the binding energy. When the electric field is larger than the Fmax value, the electron–hole distance increases, reducing the Coulomb interaction. Electric field dependence of the calculated average distance z = z e − z h , plotted in the inset of Fig. 5(b), is consistent with the one obtained by the previous estimate (Eq. (18)). Fig. 5(b) shows that for small electric fields, i.e. F ≤ 4 kV cm−1 , at a given value of PY , the exciton binding energy and the average distance are only weakly dependent upon the RM momentum p y . As the electric field increases, these quantities are determined by the competition between correlated CM and RM motions (the correlation induced by the inplane magnetic field), the effect of the electric field and the Coulomb interaction. As a result, the degeneracy for the exciton states corresponding to different p y values is resolved. For PY < 0, as expected, the states with larger RM momentum  p y  are weaker bounded, due to a larger kinetic energy p2y /2µ. For PY > 0, at intermediate fields, i.e. ex is the most important, 4 kV cm−1 ≤ F ≤ 10 kV cm−1 , the term proportional to p2y in H3D and the binding energy is lower when  p y  increases. In highfield limit, as z h rapidly ex is increases, the term linear in p y becomes dominant. Therefore, the exciton energy H3D greater for p y < 0 (d curve), leading to a diminution of the binding energy.
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4
B = 10 T c
∆E (meV)
2 d
0
2
a
4
b
0
2
4
6
8
F (kV/cm) Fig. 6. Energy shift E = E ex (PY , p y ) − E ex (PY = p y = 0) for an exciton moving perpendicularly to the magnetic field. The same curve conventions are used as in the previous figure.
The energy shift E = E ex (PY , p y )− E ex (PY = p y = 0) in a GaAs QW for B = 10 T is shown in Fig. 6 as a function of the electric field. It is interesting to observe that between 2 and 9 kV cm−1 , for this system the energy of the exciton state with PY < 0 and small  p y  becomes lower than that with PY = p y = 0. Therefore, as a function of the electric field, the ground state of the exciton exhibits a transition from PY = 0 to a state with PY = 0. A similar effect has recently been investigated experimentally [23] and theoretically [24] in symmetric coupled QWs, with [23] and without [24] applied electric field. 4. Conclusions In this paper, we study the effects of simultaneous application of electric and magnetic fields on the electronic structure of the rectangular QW. We showed that the electric field partially breaks down the degeneracy of the single particle states symmetrically positioned in p space, leading to a nonparabolic subband structure. It is found that for an exciton moving in the QW plane, the transverse magnetic field is the source of an anisotropy on the energy dispersion. In this system, at strong magnetic fields and for some electric field strengths the binding energy exhibits a maximum. Also, as a function of the electric field, induced transitions where the inplane CM momentum of the exciton changes can be obtained. Thus, controlling the values of the external fields one can enhance or reduce the energy and the spatial separation of the electron and hole in a QW. The shift of the excitonic energy levels obtained when the external applied fields vary is similar to a change in the QW width. This can be used to study these systems in regions of interest, without the need of the growth of many different samples. The assumption of a perfect confinement in the QW due to infinitely high barriers and the parabolic oneband model, used in the calculation of the single particle states,
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introduce some limitations in the validity of our results. However, it is known that this simple model is in quite good agreement with experimental data for the lowest exciton states [31], because neglecting valence band mixing underestimates the binding energy, counterbalancing in part the infinite well approximation. Also, it is expected that this model should be appropriate for large well widths in a strong magnetic field regime, when the cyclotron energy overcomes the geometric confinement energy and the effects of the electric field. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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