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Stimulated emission of terahertz acoustic phonons as a result of action of nonequilibrium phonons upon the localized exciton ensemble
Physica E 13 (2002) 321 – 324
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Stimulated emission of terahertz acoustic phonons as a result of action of nonequilibrium phonons upon the localized exciton ensemble E.E. Onishchenko ∗ , V.S. Bagaev, V.V. Zaitsev P.N. Lebedev Physical Institute RAS, Leninskii pr. 53, 119991 Moscow, Russia
Abstract The in-uence of nonequilibrium phonons created by an external source on the relaxation and lateral migration of excitons in an ultrathin CdTe=ZnTe quantum well was studied. It was found that under action of nonequilibrium phonons upon the localized exciton ensemble the transitions between localized states occur; these transitions are assumed to be accompanied by stimulated terahertz–phonon emission. The surprising fact is that the phonon e1ect reveals already for a low power of phonon generation. This indicates that a rather intense exciton relaxation in energy under the nonequilibrium phonon action occurs within the range of localized states. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Terahertz phonons; Stimulated emission; Quantum well; Photoluminescence
In the last few years, the possibility of development of terahertz acoustic phonon generators (and “phonon lasers”) based on semiconductor heterostructures is actively discussed [1–3]. There is a great number of theoretical studies but lack of experimental works. We present here likely the 9rst experimental observation of stimulated emission of terahertz acoustic phonons induced by the nonequilibrium phonon e1ect on the localized exciton ensemble. We studied the in-uence of nonequilibrium phonons on the relaxation and the lateral migration of excitons in an ultrathin CdTe=ZnTe quantum well (QW). An unusual mechanism of the lateral migration of excitons associated with the exciton tunnel transitions between localized exciton states accompanied by the stimulated phonon emission was found. ∗ Corresponding author. Tel.: +7-095-135-7941; fax: +7-095938-2251. E-mail address: [email protected] (E.E. Onishchenko).
One should note that although we use a term “ultrathin quantum well”, the ultrathin QW’s resemble the quantum dot arrays in many respects for the case of II–VI compounds. It is known that CdSe=ZnSe and CdTe=ZnTe quantum dots present “islands”—the local regions with enhanced Cd content or the local regions with enhanced thickness [4,5]. An ultrathin CdTe=ZnTe QW with a nominal thickness of CdTe layer ∼1:3 nm was grown onto the (1 0 0)GaAs substrate by MBE. The high-resolution TEM studies have shown that the ultrathin QW represents a CdTe layer of the varying local thickness [6]. The scheme of the experiment is shown in Fig. 1. The QW photoluminescence (PL) was excited by a He–Ne laser beam ( = 632:8 nm) focused into a spot (diameter ∼0:1 mm, laser power of about 1 mW). The laser radiation was modulated with a frequency of 1 kHz by a mechanical chopper Ch. For the optical generation of nonequilibrium phonons (in the processes of relaxation and nonradiative recombination of photoexcited
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 5 4 8 - 3
322
E.E. Onishchenko et al. / Physica E 13 (2002) 321 – 324
Fig. 1. Experimental geometry. Quasi-resonant PL excitation by He–Ne laser (spot diameter ∼0:1 mm, power ∼1 mW), nonequilibrium phonons were produced by an Ar laser beam. He–Ne laser radiation was modulated by a chopper Ch.
Fig. 2. Solid line—QW PL spectrum in the absence of nonequilibrium phonons. Dots—QW PL spectrum for Ar laser power 20 mW laser (spot diameter ∼0:1 mm). The region of nonequilibrium phonon generation and the region of quasi-resonant PL excitation were spaced apart.
carriers) an Ar + laser (=488 nm) was used. The Ar + laser power varied within the range 0.01–150 mW; the region of nonequilibrium phonon generation and the region of quasi-resonant PL excitation were either superimposed or spaced apart at the distances up to several millimeters. The PL spectra were analyzed by a double grating monochromator with the focal length of 0:8 m. The standard lock-in technique was used for PL detection. The measurements were carried out at temperatures 4.2 and 1:8 K. The QW PL spectra under the He–Ne laser quasi-resonant excitation are shown in Fig. 2 in the absence (solid line) and in the presence of nonequilibrium phonons (dots). The features marked by
Fig. 3. Transformation of the luminescence signal from the ultrathin QW (solid line) with increasing temperature up to 10 K and (dots) under the action of nonequilibrium phonons.
vertical dash-dotted lines are associated with the fact that the interface (h1 = 19 meV) and con9ned (h2 = 25 meV) optical phonons participate in the relaxation of excitons produced by He–Ne laser. The appearance of such kind of phonons is caused by the presence of the ultrathin CdTe layer [7]. A noticeable weakening of the features caused by the exciton relaxation assisted by con9ned and interface phonons is observed, the integral luminescence intensity being enhanced in this case. The phonon e1ect already manifests for small Ar laser power and at a distance of about 1 mm between the quasi-resonant PL excitation region and the phonon generation region. Note that an opposite behavior is observed at the temperature increase (Fig. 3). Fluctuations of the QW thickness result in appearance of a various extent of lateral exciton localization in the QW plane. The tunnel transitions between the localized exciton states with emission (absorption) of acoustic phonons are reported to be the main mechanism of lateral migration in QW’s [8]. As a rule, the higher energy states are less localized (for such a case, the time of exciton transition to another state is small compared to the recombination time). Due to the fact that the emission of optical phonons h1 and h2 serves as the main channel of a primary energy relaxation of photoexcited excitons, the predominant population of those localized exciton states occurs that are positioned from the He–Ne laser quantum by h1 and h2 . As the further exciton relaxation in energy can occur only via a considerably slower process (transition with participation of acoustic
E.E. Onishchenko et al. / Physica E 13 (2002) 321 – 324
phonons), the “excess” population of localized exciton states is realized in this energy range that is clearly pronounced in the PL spectrum (Fig. 2). The nonequilibrium phonons cause the redistribution of the occupation of localized exciton states assisting the exciton migration to the low-lying states. It is necessary to give an answer to the following questions for an explanation of the nature of the observed e1ect. First, what are the energies of phonons that can cause the transitions between localized exciton states in an ultrathin QW? Second, what is the mechanism of the exciton transfer to the low-lying localized states? Let us estimate roughly the energies of phonons which are able to cause the redistribution of the exciton states populations. In the real QW’s with thickness -uctuations the distribution of localized exciton states in energy can be considered as a Gaussian function [9]. For simplicity we assume that the above mentioned islands form a quasi-ordered system (a square lattice). Inasmuch as the tunneling probability depends exponentially on the distances between localized states, we restrict our consideration by the four nearest islands for an estimate. For the case of non-resonant excitation, the QW PL band shape is approximated by a Gaussian with the mean square deviation of about 9 meV. For the quasi-resonant excitation conditions corresponding to that shown in Fig. 2, the spectral feature associated with the exciton relaxation accompanied by con9ned LO-phonon is positioned in energy from the PL band maximum of about 2. Within the framework of our simplest model, the probability that, among the four nearest localized states, there is not a state with an energy di1erent from a localized state under consideration by the energy ¡ 4 meV (1 THz, correspondingly) is about 0.8. Therefore, the transitions between localized exciton states can be caused primarily by terahertz acoustic phonons. Let us turn to the question of a lateral migration mechanism. The situation when the excitons migrate to the lower energy localized states with the temperature increase is not unique in the low-dimensional structures. This occurs because excitons can migrate to the lower energy states via the “intermediate” (higher energy) states with the temperature increase (see Refs. [10,11] and references therein). However, this e1ect was not observed in our structure either upon the quasi-resonant excitation or upon the barrier (ZnTe) excitation.
323
As it was shown above, terahertz phonons are needed to cause considerable redistribution of the occupation of the localized exciton states. The phonon energy corresponding to the maximum of Planckian distribution is 2:5 meV at 10 K. Therefore, such a small temperature increase cannot substantially a1ect the localized exciton ensemble. The thermal redistribution of population for a small part of localized states takes place. The PL integral intensity decrease is a consequence of the fact that the population of the higher-lying states increases. It results in the nonradiative loss enhancement through the probability increase of thermal escape of excitons either to the delocalized states [12] or to the nonradiative recombination centers in the matrix material [13,14]. The latter is especially important for highly mismatched heteroepitaxial structures such as CdTe=ZnTe grown onto GaAs substrates [15]. Under the optical excitation, the spectrum of created phonons strongly di1ers from the equilibrium one. Owing to this fact, the phonon energies are suRcient to cause the tunnel transitions between localized exciton states. Since the resonantly populated states are related to the short-wavelength side of the PL band, the nearest localized states are the lower-lying ones for such states. Hence, the most probable cause of the observed e1ect is the tunnel transitions between localized states, which are accompanied by the stimulated emission of terahertz acoustic phonons. The escape of excitons into the delocalized states with a subsequent relaxation into the deeper states cannot play an important role, as we have observed the e1ect actually in the case when the states near PL band maximum are being populated, and the latter are certainly separated in energy from the delocalized states. In addition, the PL-intensity decrease should be observed in such a case, as is clear from the discussion above. It is of interest that the phonon e1ect reveals already at low power of phonon generation. This indicates that a rather intense energy relaxation of exciton under the action of nonequilibrium phonons takes place. The data on a surprisingly fast (in contrast to theoretical considerations [16,17]) carrier relaxation with emission of high-energy acoustic phonons were obtained also for InP=GaP self-assembled quantum dots [14]. It should be stressed, however, that we have two items di1ering from the situation described in Ref. [14]. First, we deal with the energy relaxation
324
E.E. Onishchenko et al. / Physica E 13 (2002) 321 – 324
over spatially separated states. Second, the exciton relaxation with the stimulated emission of terahertz acoustic phonons presumably occurs in our case. In conclusion, the e1ect of nonequilibrium acoustic phonons upon the ultrathin CdTe=ZnTe QW luminescence excited quasi-resonantly by a He–Ne laser was studied. It was established that under action of nonequilibrium phonons upon the localized exciton ensemble the transitions between localized states occur; these transitions are assumed to be accompanied by stimulated terahertz–phonon emission. Note that such an unusual mechanism of the exciton lateral migration was not observed earlier. This work was supported by the Russian Foundation for Basic Research (project no. 99-02-17183 and 00-02-17335), the FTNS program (project no. 97-1045), the Grant for the Support of Scienti9c Schools (project no. 00-15-96568), and by the Young-Scientist Support Grant of RAS no. 24.
References [1] Yu.E. Lozovik, I.V. Ovchinnikov, e-Print cond-mat= 0007162. [2] D.A. Mazurenko, A.V. Akimov, Fiz. Tverd. Tela (St. Petersburg) 43 (2001) 719 [Phys. Solid State 43 (2001) 752].
[3] I. Camps, S.S. Makler, H.M. Pastawski, L.E. Foa Torres, e-Print cond-mat=0101043. [4] M. Strassburg, Th. Deniozou, A. Ho1mann, R. Heitz, U.W. Pohl, D. Bimberg, D. Litvinov. A. Rosenauer, D. Gerthsen, S. Schwedhelm, K. Lischka, D. Schikora, Appl. Phys. Lett. 76 (2000) 685. [5] S. Mackowski, G. Karczewski, T. Wojtowicz, J. Kossut, S. Kret, A. Szczepanska, P. Dluzewski, G. Prechtl, A. Heiss, Appl. Phys. Lett. 78 (2001) 3884. [6] V.V. Zaitsev, V.S. Bagaev, E.E. Onishchenko, Fiz. Tverd. Tela (St. Petersburg) 41 (1999) 717 [Phys. Solid State 41 (1999) 647]. [7] J. Mendez, A. Pinczuk, J.P. Valladares, R.D. Feldman, R.F. Austin, Appl. Phys. Lett. 50 (1987) 1101. [8] T. Takagahara, Phys. Rev. B 31 (1985) 6552. [9] F. Yang, M. Wilkinson, E.J. Austin, K.P. O’Donnell, Phys. Rev. Lett. 70 (1993) 323. [10] S.D. Baranovskii, R. Eichmann, P. Thomas, Phys. Rev. B 58 (1998) 13081. [11] A. Polimeni, A. Patane, M. Henini, L. Eaves, P.C. Main, Phys. Rev. B 59 (1999) 5064. [12] I.N. Krivorotov, T. Chang, G.D. Gilliland, L.P. Fu, K.K. Bajaj, Phys. Rev. B 58 (1998) 10687. [13] K. Mukai, N. Ohtsuka, M. Sugawara, Appl. Phys. Lett. 70 (1997) 2416. [14] I.V. Ignatiev, I.E. Kozin, V.G. Davydov, S.V. Nair, J.-S. Lee, H.-W. Ren, S. Sugou, Y. Masumoto, Phys. Rev. B 63 (2001) 075316. [15] V.S. Bagaev, V.V. Zaitsev, E.E. Onishchenko, Yu.G. Sadofyev, J. Crystal. Growth 214=215 (2000) 250. [16] U. Bockelmann, G. Bastard, Phys. Rev. B 42 (1990) 8947. [17] H. Benisty, C.M. Sotomayor-Torres, C. Weisbuch, Phys. Rev. B 42 (1991) 10945.
www.elsevier.com/locate/physe
Stimulated emission of terahertz acoustic phonons as a result of action of nonequilibrium phonons upon the localized exciton ensemble E.E. Onishchenko ∗ , V.S. Bagaev, V.V. Zaitsev P.N. Lebedev Physical Institute RAS, Leninskii pr. 53, 119991 Moscow, Russia
Abstract The in-uence of nonequilibrium phonons created by an external source on the relaxation and lateral migration of excitons in an ultrathin CdTe=ZnTe quantum well was studied. It was found that under action of nonequilibrium phonons upon the localized exciton ensemble the transitions between localized states occur; these transitions are assumed to be accompanied by stimulated terahertz–phonon emission. The surprising fact is that the phonon e1ect reveals already for a low power of phonon generation. This indicates that a rather intense exciton relaxation in energy under the nonequilibrium phonon action occurs within the range of localized states. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Terahertz phonons; Stimulated emission; Quantum well; Photoluminescence
In the last few years, the possibility of development of terahertz acoustic phonon generators (and “phonon lasers”) based on semiconductor heterostructures is actively discussed [1–3]. There is a great number of theoretical studies but lack of experimental works. We present here likely the 9rst experimental observation of stimulated emission of terahertz acoustic phonons induced by the nonequilibrium phonon e1ect on the localized exciton ensemble. We studied the in-uence of nonequilibrium phonons on the relaxation and the lateral migration of excitons in an ultrathin CdTe=ZnTe quantum well (QW). An unusual mechanism of the lateral migration of excitons associated with the exciton tunnel transitions between localized exciton states accompanied by the stimulated phonon emission was found. ∗ Corresponding author. Tel.: +7-095-135-7941; fax: +7-095938-2251. E-mail address: [email protected] (E.E. Onishchenko).
One should note that although we use a term “ultrathin quantum well”, the ultrathin QW’s resemble the quantum dot arrays in many respects for the case of II–VI compounds. It is known that CdSe=ZnSe and CdTe=ZnTe quantum dots present “islands”—the local regions with enhanced Cd content or the local regions with enhanced thickness [4,5]. An ultrathin CdTe=ZnTe QW with a nominal thickness of CdTe layer ∼1:3 nm was grown onto the (1 0 0)GaAs substrate by MBE. The high-resolution TEM studies have shown that the ultrathin QW represents a CdTe layer of the varying local thickness [6]. The scheme of the experiment is shown in Fig. 1. The QW photoluminescence (PL) was excited by a He–Ne laser beam ( = 632:8 nm) focused into a spot (diameter ∼0:1 mm, laser power of about 1 mW). The laser radiation was modulated with a frequency of 1 kHz by a mechanical chopper Ch. For the optical generation of nonequilibrium phonons (in the processes of relaxation and nonradiative recombination of photoexcited
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 5 4 8 - 3
322
E.E. Onishchenko et al. / Physica E 13 (2002) 321 – 324
Fig. 1. Experimental geometry. Quasi-resonant PL excitation by He–Ne laser (spot diameter ∼0:1 mm, power ∼1 mW), nonequilibrium phonons were produced by an Ar laser beam. He–Ne laser radiation was modulated by a chopper Ch.
Fig. 2. Solid line—QW PL spectrum in the absence of nonequilibrium phonons. Dots—QW PL spectrum for Ar laser power 20 mW laser (spot diameter ∼0:1 mm). The region of nonequilibrium phonon generation and the region of quasi-resonant PL excitation were spaced apart.
carriers) an Ar + laser (=488 nm) was used. The Ar + laser power varied within the range 0.01–150 mW; the region of nonequilibrium phonon generation and the region of quasi-resonant PL excitation were either superimposed or spaced apart at the distances up to several millimeters. The PL spectra were analyzed by a double grating monochromator with the focal length of 0:8 m. The standard lock-in technique was used for PL detection. The measurements were carried out at temperatures 4.2 and 1:8 K. The QW PL spectra under the He–Ne laser quasi-resonant excitation are shown in Fig. 2 in the absence (solid line) and in the presence of nonequilibrium phonons (dots). The features marked by
Fig. 3. Transformation of the luminescence signal from the ultrathin QW (solid line) with increasing temperature up to 10 K and (dots) under the action of nonequilibrium phonons.
vertical dash-dotted lines are associated with the fact that the interface (h1 = 19 meV) and con9ned (h2 = 25 meV) optical phonons participate in the relaxation of excitons produced by He–Ne laser. The appearance of such kind of phonons is caused by the presence of the ultrathin CdTe layer [7]. A noticeable weakening of the features caused by the exciton relaxation assisted by con9ned and interface phonons is observed, the integral luminescence intensity being enhanced in this case. The phonon e1ect already manifests for small Ar laser power and at a distance of about 1 mm between the quasi-resonant PL excitation region and the phonon generation region. Note that an opposite behavior is observed at the temperature increase (Fig. 3). Fluctuations of the QW thickness result in appearance of a various extent of lateral exciton localization in the QW plane. The tunnel transitions between the localized exciton states with emission (absorption) of acoustic phonons are reported to be the main mechanism of lateral migration in QW’s [8]. As a rule, the higher energy states are less localized (for such a case, the time of exciton transition to another state is small compared to the recombination time). Due to the fact that the emission of optical phonons h1 and h2 serves as the main channel of a primary energy relaxation of photoexcited excitons, the predominant population of those localized exciton states occurs that are positioned from the He–Ne laser quantum by h1 and h2 . As the further exciton relaxation in energy can occur only via a considerably slower process (transition with participation of acoustic
E.E. Onishchenko et al. / Physica E 13 (2002) 321 – 324
phonons), the “excess” population of localized exciton states is realized in this energy range that is clearly pronounced in the PL spectrum (Fig. 2). The nonequilibrium phonons cause the redistribution of the occupation of localized exciton states assisting the exciton migration to the low-lying states. It is necessary to give an answer to the following questions for an explanation of the nature of the observed e1ect. First, what are the energies of phonons that can cause the transitions between localized exciton states in an ultrathin QW? Second, what is the mechanism of the exciton transfer to the low-lying localized states? Let us estimate roughly the energies of phonons which are able to cause the redistribution of the exciton states populations. In the real QW’s with thickness -uctuations the distribution of localized exciton states in energy can be considered as a Gaussian function [9]. For simplicity we assume that the above mentioned islands form a quasi-ordered system (a square lattice). Inasmuch as the tunneling probability depends exponentially on the distances between localized states, we restrict our consideration by the four nearest islands for an estimate. For the case of non-resonant excitation, the QW PL band shape is approximated by a Gaussian with the mean square deviation of about 9 meV. For the quasi-resonant excitation conditions corresponding to that shown in Fig. 2, the spectral feature associated with the exciton relaxation accompanied by con9ned LO-phonon is positioned in energy from the PL band maximum of about 2. Within the framework of our simplest model, the probability that, among the four nearest localized states, there is not a state with an energy di1erent from a localized state under consideration by the energy ¡ 4 meV (1 THz, correspondingly) is about 0.8. Therefore, the transitions between localized exciton states can be caused primarily by terahertz acoustic phonons. Let us turn to the question of a lateral migration mechanism. The situation when the excitons migrate to the lower energy localized states with the temperature increase is not unique in the low-dimensional structures. This occurs because excitons can migrate to the lower energy states via the “intermediate” (higher energy) states with the temperature increase (see Refs. [10,11] and references therein). However, this e1ect was not observed in our structure either upon the quasi-resonant excitation or upon the barrier (ZnTe) excitation.
323
As it was shown above, terahertz phonons are needed to cause considerable redistribution of the occupation of the localized exciton states. The phonon energy corresponding to the maximum of Planckian distribution is 2:5 meV at 10 K. Therefore, such a small temperature increase cannot substantially a1ect the localized exciton ensemble. The thermal redistribution of population for a small part of localized states takes place. The PL integral intensity decrease is a consequence of the fact that the population of the higher-lying states increases. It results in the nonradiative loss enhancement through the probability increase of thermal escape of excitons either to the delocalized states [12] or to the nonradiative recombination centers in the matrix material [13,14]. The latter is especially important for highly mismatched heteroepitaxial structures such as CdTe=ZnTe grown onto GaAs substrates [15]. Under the optical excitation, the spectrum of created phonons strongly di1ers from the equilibrium one. Owing to this fact, the phonon energies are suRcient to cause the tunnel transitions between localized exciton states. Since the resonantly populated states are related to the short-wavelength side of the PL band, the nearest localized states are the lower-lying ones for such states. Hence, the most probable cause of the observed e1ect is the tunnel transitions between localized states, which are accompanied by the stimulated emission of terahertz acoustic phonons. The escape of excitons into the delocalized states with a subsequent relaxation into the deeper states cannot play an important role, as we have observed the e1ect actually in the case when the states near PL band maximum are being populated, and the latter are certainly separated in energy from the delocalized states. In addition, the PL-intensity decrease should be observed in such a case, as is clear from the discussion above. It is of interest that the phonon e1ect reveals already at low power of phonon generation. This indicates that a rather intense energy relaxation of exciton under the action of nonequilibrium phonons takes place. The data on a surprisingly fast (in contrast to theoretical considerations [16,17]) carrier relaxation with emission of high-energy acoustic phonons were obtained also for InP=GaP self-assembled quantum dots [14]. It should be stressed, however, that we have two items di1ering from the situation described in Ref. [14]. First, we deal with the energy relaxation
324
E.E. Onishchenko et al. / Physica E 13 (2002) 321 – 324
over spatially separated states. Second, the exciton relaxation with the stimulated emission of terahertz acoustic phonons presumably occurs in our case. In conclusion, the e1ect of nonequilibrium acoustic phonons upon the ultrathin CdTe=ZnTe QW luminescence excited quasi-resonantly by a He–Ne laser was studied. It was established that under action of nonequilibrium phonons upon the localized exciton ensemble the transitions between localized states occur; these transitions are assumed to be accompanied by stimulated terahertz–phonon emission. Note that such an unusual mechanism of the exciton lateral migration was not observed earlier. This work was supported by the Russian Foundation for Basic Research (project no. 99-02-17183 and 00-02-17335), the FTNS program (project no. 97-1045), the Grant for the Support of Scienti9c Schools (project no. 00-15-96568), and by the Young-Scientist Support Grant of RAS no. 24.
References [1] Yu.E. Lozovik, I.V. Ovchinnikov, e-Print cond-mat= 0007162. [2] D.A. Mazurenko, A.V. Akimov, Fiz. Tverd. Tela (St. Petersburg) 43 (2001) 719 [Phys. Solid State 43 (2001) 752].
[3] I. Camps, S.S. Makler, H.M. Pastawski, L.E. Foa Torres, e-Print cond-mat=0101043. [4] M. Strassburg, Th. Deniozou, A. Ho1mann, R. Heitz, U.W. Pohl, D. Bimberg, D. Litvinov. A. Rosenauer, D. Gerthsen, S. Schwedhelm, K. Lischka, D. Schikora, Appl. Phys. Lett. 76 (2000) 685. [5] S. Mackowski, G. Karczewski, T. Wojtowicz, J. Kossut, S. Kret, A. Szczepanska, P. Dluzewski, G. Prechtl, A. Heiss, Appl. Phys. Lett. 78 (2001) 3884. [6] V.V. Zaitsev, V.S. Bagaev, E.E. Onishchenko, Fiz. Tverd. Tela (St. Petersburg) 41 (1999) 717 [Phys. Solid State 41 (1999) 647]. [7] J. Mendez, A. Pinczuk, J.P. Valladares, R.D. Feldman, R.F. Austin, Appl. Phys. Lett. 50 (1987) 1101. [8] T. Takagahara, Phys. Rev. B 31 (1985) 6552. [9] F. Yang, M. Wilkinson, E.J. Austin, K.P. O’Donnell, Phys. Rev. Lett. 70 (1993) 323. [10] S.D. Baranovskii, R. Eichmann, P. Thomas, Phys. Rev. B 58 (1998) 13081. [11] A. Polimeni, A. Patane, M. Henini, L. Eaves, P.C. Main, Phys. Rev. B 59 (1999) 5064. [12] I.N. Krivorotov, T. Chang, G.D. Gilliland, L.P. Fu, K.K. Bajaj, Phys. Rev. B 58 (1998) 10687. [13] K. Mukai, N. Ohtsuka, M. Sugawara, Appl. Phys. Lett. 70 (1997) 2416. [14] I.V. Ignatiev, I.E. Kozin, V.G. Davydov, S.V. Nair, J.-S. Lee, H.-W. Ren, S. Sugou, Y. Masumoto, Phys. Rev. B 63 (2001) 075316. [15] V.S. Bagaev, V.V. Zaitsev, E.E. Onishchenko, Yu.G. Sadofyev, J. Crystal. Growth 214=215 (2000) 250. [16] U. Bockelmann, G. Bastard, Phys. Rev. B 42 (1990) 8947. [17] H. Benisty, C.M. Sotomayor-Torres, C. Weisbuch, Phys. Rev. B 42 (1991) 10945.