- Email: [email protected]
Relaxation dynamics of ferromagnetic domains in (Cd,Mn)Te quantum wells
ARTICLE IN PRESS
Physica E 32 (2006) 454–457 www.elsevier.com/locate/physe
Relaxation dynamics of ferromagnetic domains in (Cd,Mn)Te quantum wells P. Kossackia,b,, D. Ferrandb, M. Gorycaa, M. Nawrockia, W. Pacuskia,b, W. Mas´ lanaa,b, S. Tatarenkob, J. Cibertc a Institute of Experimental Physics, Warsaw University, Hoza 69, 00-681 Warsaw, Poland Joined group ‘‘Nanophysique et semiconducteurs’’, CNRS/CEA/Universite´ Joseph Fourier-Grenoble, Laboratoire de Spectrome´trie Physique, BP 87, 38402 Saint Martin d’He`res cedex, France c Laboratoire Louis Ne´el, CNRS, BP166, 38042 Grenoble cedex 9, France
b
Available online 30 January 2006
Abstract We present a magneto-optical study of the magnetization dynamics in ferromagnetic quantum wells in the time range down to 20 ns. We use a small electromagnet coil to produce short pulses of magnetic field. The magnetization relaxation after the pulse (observed through PL polarization) is faster than 20 ns in the paramagnetic state. Decreasing the temperature below the Curie temperature T C results in an increase of the relaxation time up to 2 ms. This relaxation corresponds to the dynamics of the ferromagnetic domains. The mobility of the domains increases with temperature. r 2006 Elsevier B.V. All rights reserved. PACS: 71.35.Pq; 71.70.Gm; 75.50.Dd; 78.55.Et; 78.67.De; 85.75.d Keywords: Quantum wells; Ferromagnetic semiconductors; Magnetization relaxation; Ferromagnetic domains
Modulation-doped (Cd,Mn)Te/(Cd,Mg,Zn)Te quantum wells (QW) form a model system for the study of ferromagnetism induced by two-dimensional carriers. Much of our understanding of this system has been achieved through magneto-optical spectroscopy [1,2]. In particular, the local spontaneous magnetization was deduced from the zero-field splitting of the QW photoluminescence line, which appears below the Curie temperature T C . The degree of circular polarization of both lines was used as a measure of the total magnetization [3]. It was shown previously, from a study of the circular polarization memory, that in typical samples magnetic domains exist, corresponding to the two spin orientations normal to the QW, and that these domains are stable over the exciton life-time [4]. On the other hand, field scans
Corresponding author. Institute of Experimental Physics, Warsaw University, Hoza 69, 00-681 Warsaw, Poland. Tel.: +48 22 55 32 217; fax: +48 22 62 19 712. E-mail address: [email protected] (P. Kossacki).
1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.12.087
show that the coercive field is negligibly small, which implies a high mobility of the magnetic domains. We present a direct magneto-optical study of the magnetization dynamics in ferromagnetic QWs in the time range down to 20 ns, at vanishing or small values of the applied magnetic field. This is achieved by applying short magnetic pulses produced with a small coil, and using the giant Zeeman effect to detect the resulting change of the magnetization. The samples, grown by molecular beam epitaxy on Cd0:88 Zn0:12 Te ð0 0 1Þ substrates, contain a single 8 nm wide p-type Cd1x Mnx Te QW. The barriers are made of Cd1yz Mgy Znz Te, in which the Mg content (y ¼ 0:25–0.28) determines the valence band offset, while the presence of Zn (z ¼ 0:08–0.07) ensures a good lattice match to the substrate. The barriers are modulation doped by nitrogen. The distance between the QW and the doping layer (20 nm) results in a hole density up to 3 1011 cm2 . The samples were mounted strain free and placed in an optical cryostat. Short pulses of magnetic field were produced with a magnetic coil mounted at the surface of
ARTICLE IN PRESS P. Kossacki et al. / Physica E 32 (2006) 454–457
the sample. The illumination with the laser beam and the collection of the photoluminescence (PL) signal were performed along the axis of the coil (Faraday configuration). The size of the coil (about 0.5 mm in diameter) assured small inductance of it and allowed us to obtain short pulses of the magnetic field, with rise and fall times of about 10 ns. A 2 A current produced a magnetic field of about 40 mT, as shown below. We analyzed the evolution in time of the PL during and after the pulse, with a resolution down to 10 ns. A typical PL spectrum measured on a paramagnetic p-type (Cd,Mn)Te QW is shown in Fig. 1a. It consists of a single line related to the charged exciton Xþ . Applying a magnetic field results in a Zeeman shift of the line opposite for both circular polarizations. Thus the intensity of the PL signal detected on the high energy side of the line increases in s polarization and decreases in sþ . It is opposite on the other side of the line. Figs. 1a, b show examples of temporal profiles of the variation of the PL intensity recorded at two wavelengths during and after the pulse of magnetic field. The relaxation dynamics can be studied by analyzing the difference of PL intensities in the two circular polarizations after the end of the magnetic field pulse [5]. The PL spectra measured on a ferromagnetic sample (4% of Mn) at temperature below T C are displayed in Fig. 2b. The thick line represents the zero-field spectrum. It clearly contains two lines, which have been related to the
(b)
(c)
Photoluminescence
σ+ σ-
3
σσ+
field pulse
2 0
Photoluminescence
5
10
0 time [µs]
10
(a)
4 3
σ+
σ-
2 1 761
762
763
764
wavelength [nm] Fig. 1. (a) Typical PL spectra of a paramagnetic sample (0.6% Mn). The vertical lines show wavelengths for which the temporal profiles shown in the upper part were measured in pulsed magnetic field: (b) 762.1 nm; and (c) 762.8 nm.
455
two Zeeman components in the presence of a spontaneous magnetization normal to the QW [1]. The splitting is proportional to the internal magnetization within domains. The overall spectrum is not circularly polarized, because it averages over domains with the two possible orientations normal to the QW [3]. Applying a small magnetic field favors one orientation of the domains. The high-energy line from these domains rapidly dominates the sþ polarized spectrum, while their low-energy line dominates the s spectrum. The lines from minority domains exhibit opposite polarizations and their intensity rapidly decreases with field. Example spectra are shown by thin solid lines in Fig. 2b. From the polarization degree of each line, it is possible to obtain the overall magnetization in the QW [3]. Figs. 2a, c show examples of the variation of the PL intensity recorded at two wavelengths during and after the pulse of magnetic field. The measurement was done in zero DC field, in two circular polarizations. The PL signal saturates after a few microseconds. These values obtained at long time during and after the pulse are shown by symbols in Fig. 2b, and compared to DC spectra and zerofield spectra, respectively. A good agreement is found with the zero-field spectrum, on one hand, and with the polarized spectra obtained with a DC field B ¼ 20 mT, on the other hand. We conclude that the measured signal is due to the evolution of the magnetic domains, as previously discussed for the DC spectra [3]. As a by-product, this allows us to calibrate the intensity of the magnetic field produced by the coil; it is of the same order as the saturation field measured in Ref. [3]. The relaxation dynamics was studied by analyzing the difference of PL intensities in the two circular polarizations after the end of the magnetic field pulse, i.e., with no applied field. Measurements were done at different temperatures below and above T C (equal 2.5 K for studied sample). Above T C , in the paramagnetic state, a single PL line is observed at zero field, and the signal induced by the magnetic pulse is related to the Zeeman splitting of this line. Its intensity was found to be significantly smaller than in the ferromagnetic phase, with a relaxation faster than 20 ns. We ascribe such a finding to the effect of the zerofield spin decoherence time T 2 [6]; its value is characteristic for bulk (Cd,Mn)Te material with 4% Mn [6,7]. Slower dynamics is observed below T C for the formation and reorientation of ferromagnetic domains after switching off the magnetic field (Fig. 3). The decay is not monoexponential, with the longest components as long as a few microseconds. This dynamics of the magnetic domains was also compared to the relaxation of the magnetization in the same (Cd,Mn)Te QW in the paramagnetic phase, at the same temperature. The transition from the ferromagnetic to the paramagnetic phase at constant temperature was achieved by illumination of the sample with light of energy above the energy gap of barrier material. Such
ARTICLE IN PRESS P. Kossacki et al. / Physica E 32 (2006) 454–457
456
PL intensity [arb.u.]
σ+
current [A] pulse 1
1000
0
750 σ-
500
0.02T
0T σ-
0.02T
250 σ+ 0
c
a
σ+
B=0.04T
0
σ(a)
10 20 30 40 time [µs]
(b)
0.04T 724
726
728 730 732 wave length [nm]
734
(c) 0
10
20 30 40 time [µs]
Photoluminescence Polarization [arb.u.]
Fig. 2. (b) Typical PL spectra measured at 1.4 K in zero (thick line) and small (thin lines) DC field. The vertical lines show wavelengths for which the temporal profiles were measured in pulsed magnetic field: (a) 725.5 nm; and (c) 730.5 nm. Symbols in (b) denote PL intensity value measures at long delay after (squares) and during the magnetic pulse, either in sþ (circles) or s (triangles) polarization. The profile of the current pulse is shown by the dashed line in (a).
1.35 K 1.59 K 4.2 K 1.5K no holes 10
10
τ=2.1µs
τ=20ns 1
1
0
200 time [ns]
400
0
2000 time [ns]
4000
Fig. 3. Temporal profiles of the circular polarization of the PL following the pulse of magnetic field, at different temperatures as indicated. Thin solid lines show an exponential decay. The T C was 2.5 K.
an illumination depletes the hole gas: then the empty (Cd,Mn)Te QW is paramagnetic [2], so that the measured signal is related to the Zeeman shift of the PL line. A typical relaxation trace is also shown in Fig. 3: the relaxation time is as short as in the paramagnetic phase above T C . The fast relaxation time previously observed in bulk paramagnetic (Cd,Mn)Te was observed to significantly slow down when applying a magnetic field [6]. The same mechanism could contribute to the slowing down of the relaxation in the ferromagnetic phase, since the polarization of the carriers induces an exchange field acting on the Mn spins. However, when applying a DC field of the same order of magnitude (0.2 T), we did not observe such an increase of the relaxation time. Also, as already noticed, the amplitude of the signal in the ferromagnetic phase agrees well with the DC spectra. Therefore, we interpret the long relaxation profiles observed in the
ferromagnetic phase as resulting from the evolution of the domains. As it is shown in Fig. 3, the relaxation begins with a fast decay, faster than 100 ns, but it also contains slower components, in the microsecond range. Such a behavior might be due to the presence of different pinning centers, preventing the motion of domain walls and making their reorientation difficult. The relaxation trace would then be essentially inhomogeneous, with the initial part of the relaxation involving the most mobile domain walls, and slower processes playing an important role in the long time tail [8,9]. Such an interpretation is supported by the observed decrease of the characteristic times when increasing the temperature. Even in the ferromagnetic phase, increasing the temperature decreases magnetization within the domains and hence decreases the energy barrier necessary to change the surface of the domains. Actually, our knowledge of the nature of domains in this system is
ARTICLE IN PRESS P. Kossacki et al. / Physica E 32 (2006) 454–457
still not sufficient, but the role of disorder is thought to be crucial [10]. One should note that, even for the lowest temperatures, the relaxation time is shorter than 10 ms, and that under all experimental conditions the remanent magnetization vanishes totally after a delay time no longer than 20 ms. This delay is much shorter than an acquisition time in optical measurements performed previously. This accounts for the very small coercive field observed so far [3]. In conclusion, from the analysis of the dynamics of the magnetization decay measured after a pulse of magnetic field the relaxation was found to be shorter than 20 ns in the paramagnetic state. Decreasing the temperature below T C results in an increase of the relaxation time up to 2 ms. This relaxation corresponds to a reorganization of ferromagnetic domains with domain mobility increasing with temperature.
457
This work was partially supported by Polish Committee for Scientific Research (Grants 2P03B 002 25 and PBZKBN-044/P03/2001) and Polonium Program.
References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10]
A. Haury, et al., Phys. Rev. Lett. 79 (1997) 511. H. Boukari, et al., Phys. Rev. Lett. 88 (2002) 207204. P. Kossacki, et al., Physica E 6 (2000) 709. P. Kossacki, et al., Physica E 12 (2002) 344. M. Goryca, D. Ferrand, P. Kossacki, M. Nawrocki, W. Pacuski, W. Mas´ lana, S. Tatarenko, J. Cibert, Phys. Stat. Sol. (b) (2006), to be published. D. Scalbert, et al., Solid State Comm. 66 (1988) 571. H.A. Sayad, S.M. Bhagat, Phys. Rev. B 31 (1985) 591. R.D. Kirby, et al., Phys. Rev. B 49 (1994) 10810. U. Nowak, et al., Phys. Rev. B 56 (1997) 8143. D. Kechrakos, et al., Phys. Rev. Lett. 94 (2005) 127201.
Physica E 32 (2006) 454–457 www.elsevier.com/locate/physe
Relaxation dynamics of ferromagnetic domains in (Cd,Mn)Te quantum wells P. Kossackia,b,, D. Ferrandb, M. Gorycaa, M. Nawrockia, W. Pacuskia,b, W. Mas´ lanaa,b, S. Tatarenkob, J. Cibertc a Institute of Experimental Physics, Warsaw University, Hoza 69, 00-681 Warsaw, Poland Joined group ‘‘Nanophysique et semiconducteurs’’, CNRS/CEA/Universite´ Joseph Fourier-Grenoble, Laboratoire de Spectrome´trie Physique, BP 87, 38402 Saint Martin d’He`res cedex, France c Laboratoire Louis Ne´el, CNRS, BP166, 38042 Grenoble cedex 9, France
b
Available online 30 January 2006
Abstract We present a magneto-optical study of the magnetization dynamics in ferromagnetic quantum wells in the time range down to 20 ns. We use a small electromagnet coil to produce short pulses of magnetic field. The magnetization relaxation after the pulse (observed through PL polarization) is faster than 20 ns in the paramagnetic state. Decreasing the temperature below the Curie temperature T C results in an increase of the relaxation time up to 2 ms. This relaxation corresponds to the dynamics of the ferromagnetic domains. The mobility of the domains increases with temperature. r 2006 Elsevier B.V. All rights reserved. PACS: 71.35.Pq; 71.70.Gm; 75.50.Dd; 78.55.Et; 78.67.De; 85.75.d Keywords: Quantum wells; Ferromagnetic semiconductors; Magnetization relaxation; Ferromagnetic domains
Modulation-doped (Cd,Mn)Te/(Cd,Mg,Zn)Te quantum wells (QW) form a model system for the study of ferromagnetism induced by two-dimensional carriers. Much of our understanding of this system has been achieved through magneto-optical spectroscopy [1,2]. In particular, the local spontaneous magnetization was deduced from the zero-field splitting of the QW photoluminescence line, which appears below the Curie temperature T C . The degree of circular polarization of both lines was used as a measure of the total magnetization [3]. It was shown previously, from a study of the circular polarization memory, that in typical samples magnetic domains exist, corresponding to the two spin orientations normal to the QW, and that these domains are stable over the exciton life-time [4]. On the other hand, field scans
Corresponding author. Institute of Experimental Physics, Warsaw University, Hoza 69, 00-681 Warsaw, Poland. Tel.: +48 22 55 32 217; fax: +48 22 62 19 712. E-mail address: [email protected] (P. Kossacki).
1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.12.087
show that the coercive field is negligibly small, which implies a high mobility of the magnetic domains. We present a direct magneto-optical study of the magnetization dynamics in ferromagnetic QWs in the time range down to 20 ns, at vanishing or small values of the applied magnetic field. This is achieved by applying short magnetic pulses produced with a small coil, and using the giant Zeeman effect to detect the resulting change of the magnetization. The samples, grown by molecular beam epitaxy on Cd0:88 Zn0:12 Te ð0 0 1Þ substrates, contain a single 8 nm wide p-type Cd1x Mnx Te QW. The barriers are made of Cd1yz Mgy Znz Te, in which the Mg content (y ¼ 0:25–0.28) determines the valence band offset, while the presence of Zn (z ¼ 0:08–0.07) ensures a good lattice match to the substrate. The barriers are modulation doped by nitrogen. The distance between the QW and the doping layer (20 nm) results in a hole density up to 3 1011 cm2 . The samples were mounted strain free and placed in an optical cryostat. Short pulses of magnetic field were produced with a magnetic coil mounted at the surface of
ARTICLE IN PRESS P. Kossacki et al. / Physica E 32 (2006) 454–457
the sample. The illumination with the laser beam and the collection of the photoluminescence (PL) signal were performed along the axis of the coil (Faraday configuration). The size of the coil (about 0.5 mm in diameter) assured small inductance of it and allowed us to obtain short pulses of the magnetic field, with rise and fall times of about 10 ns. A 2 A current produced a magnetic field of about 40 mT, as shown below. We analyzed the evolution in time of the PL during and after the pulse, with a resolution down to 10 ns. A typical PL spectrum measured on a paramagnetic p-type (Cd,Mn)Te QW is shown in Fig. 1a. It consists of a single line related to the charged exciton Xþ . Applying a magnetic field results in a Zeeman shift of the line opposite for both circular polarizations. Thus the intensity of the PL signal detected on the high energy side of the line increases in s polarization and decreases in sþ . It is opposite on the other side of the line. Figs. 1a, b show examples of temporal profiles of the variation of the PL intensity recorded at two wavelengths during and after the pulse of magnetic field. The relaxation dynamics can be studied by analyzing the difference of PL intensities in the two circular polarizations after the end of the magnetic field pulse [5]. The PL spectra measured on a ferromagnetic sample (4% of Mn) at temperature below T C are displayed in Fig. 2b. The thick line represents the zero-field spectrum. It clearly contains two lines, which have been related to the
(b)
(c)
Photoluminescence
σ+ σ-
3
σσ+
field pulse
2 0
Photoluminescence
5
10
0 time [µs]
10
(a)
4 3
σ+
σ-
2 1 761
762
763
764
wavelength [nm] Fig. 1. (a) Typical PL spectra of a paramagnetic sample (0.6% Mn). The vertical lines show wavelengths for which the temporal profiles shown in the upper part were measured in pulsed magnetic field: (b) 762.1 nm; and (c) 762.8 nm.
455
two Zeeman components in the presence of a spontaneous magnetization normal to the QW [1]. The splitting is proportional to the internal magnetization within domains. The overall spectrum is not circularly polarized, because it averages over domains with the two possible orientations normal to the QW [3]. Applying a small magnetic field favors one orientation of the domains. The high-energy line from these domains rapidly dominates the sþ polarized spectrum, while their low-energy line dominates the s spectrum. The lines from minority domains exhibit opposite polarizations and their intensity rapidly decreases with field. Example spectra are shown by thin solid lines in Fig. 2b. From the polarization degree of each line, it is possible to obtain the overall magnetization in the QW [3]. Figs. 2a, c show examples of the variation of the PL intensity recorded at two wavelengths during and after the pulse of magnetic field. The measurement was done in zero DC field, in two circular polarizations. The PL signal saturates after a few microseconds. These values obtained at long time during and after the pulse are shown by symbols in Fig. 2b, and compared to DC spectra and zerofield spectra, respectively. A good agreement is found with the zero-field spectrum, on one hand, and with the polarized spectra obtained with a DC field B ¼ 20 mT, on the other hand. We conclude that the measured signal is due to the evolution of the magnetic domains, as previously discussed for the DC spectra [3]. As a by-product, this allows us to calibrate the intensity of the magnetic field produced by the coil; it is of the same order as the saturation field measured in Ref. [3]. The relaxation dynamics was studied by analyzing the difference of PL intensities in the two circular polarizations after the end of the magnetic field pulse, i.e., with no applied field. Measurements were done at different temperatures below and above T C (equal 2.5 K for studied sample). Above T C , in the paramagnetic state, a single PL line is observed at zero field, and the signal induced by the magnetic pulse is related to the Zeeman splitting of this line. Its intensity was found to be significantly smaller than in the ferromagnetic phase, with a relaxation faster than 20 ns. We ascribe such a finding to the effect of the zerofield spin decoherence time T 2 [6]; its value is characteristic for bulk (Cd,Mn)Te material with 4% Mn [6,7]. Slower dynamics is observed below T C for the formation and reorientation of ferromagnetic domains after switching off the magnetic field (Fig. 3). The decay is not monoexponential, with the longest components as long as a few microseconds. This dynamics of the magnetic domains was also compared to the relaxation of the magnetization in the same (Cd,Mn)Te QW in the paramagnetic phase, at the same temperature. The transition from the ferromagnetic to the paramagnetic phase at constant temperature was achieved by illumination of the sample with light of energy above the energy gap of barrier material. Such
ARTICLE IN PRESS P. Kossacki et al. / Physica E 32 (2006) 454–457
456
PL intensity [arb.u.]
σ+
current [A] pulse 1
1000
0
750 σ-
500
0.02T
0T σ-
0.02T
250 σ+ 0
c
a
σ+
B=0.04T
0
σ(a)
10 20 30 40 time [µs]
(b)
0.04T 724
726
728 730 732 wave length [nm]
734
(c) 0
10
20 30 40 time [µs]
Photoluminescence Polarization [arb.u.]
Fig. 2. (b) Typical PL spectra measured at 1.4 K in zero (thick line) and small (thin lines) DC field. The vertical lines show wavelengths for which the temporal profiles were measured in pulsed magnetic field: (a) 725.5 nm; and (c) 730.5 nm. Symbols in (b) denote PL intensity value measures at long delay after (squares) and during the magnetic pulse, either in sþ (circles) or s (triangles) polarization. The profile of the current pulse is shown by the dashed line in (a).
1.35 K 1.59 K 4.2 K 1.5K no holes 10
10
τ=2.1µs
τ=20ns 1
1
0
200 time [ns]
400
0
2000 time [ns]
4000
Fig. 3. Temporal profiles of the circular polarization of the PL following the pulse of magnetic field, at different temperatures as indicated. Thin solid lines show an exponential decay. The T C was 2.5 K.
an illumination depletes the hole gas: then the empty (Cd,Mn)Te QW is paramagnetic [2], so that the measured signal is related to the Zeeman shift of the PL line. A typical relaxation trace is also shown in Fig. 3: the relaxation time is as short as in the paramagnetic phase above T C . The fast relaxation time previously observed in bulk paramagnetic (Cd,Mn)Te was observed to significantly slow down when applying a magnetic field [6]. The same mechanism could contribute to the slowing down of the relaxation in the ferromagnetic phase, since the polarization of the carriers induces an exchange field acting on the Mn spins. However, when applying a DC field of the same order of magnitude (0.2 T), we did not observe such an increase of the relaxation time. Also, as already noticed, the amplitude of the signal in the ferromagnetic phase agrees well with the DC spectra. Therefore, we interpret the long relaxation profiles observed in the
ferromagnetic phase as resulting from the evolution of the domains. As it is shown in Fig. 3, the relaxation begins with a fast decay, faster than 100 ns, but it also contains slower components, in the microsecond range. Such a behavior might be due to the presence of different pinning centers, preventing the motion of domain walls and making their reorientation difficult. The relaxation trace would then be essentially inhomogeneous, with the initial part of the relaxation involving the most mobile domain walls, and slower processes playing an important role in the long time tail [8,9]. Such an interpretation is supported by the observed decrease of the characteristic times when increasing the temperature. Even in the ferromagnetic phase, increasing the temperature decreases magnetization within the domains and hence decreases the energy barrier necessary to change the surface of the domains. Actually, our knowledge of the nature of domains in this system is
ARTICLE IN PRESS P. Kossacki et al. / Physica E 32 (2006) 454–457
still not sufficient, but the role of disorder is thought to be crucial [10]. One should note that, even for the lowest temperatures, the relaxation time is shorter than 10 ms, and that under all experimental conditions the remanent magnetization vanishes totally after a delay time no longer than 20 ms. This delay is much shorter than an acquisition time in optical measurements performed previously. This accounts for the very small coercive field observed so far [3]. In conclusion, from the analysis of the dynamics of the magnetization decay measured after a pulse of magnetic field the relaxation was found to be shorter than 20 ns in the paramagnetic state. Decreasing the temperature below T C results in an increase of the relaxation time up to 2 ms. This relaxation corresponds to a reorganization of ferromagnetic domains with domain mobility increasing with temperature.
457
This work was partially supported by Polish Committee for Scientific Research (Grants 2P03B 002 25 and PBZKBN-044/P03/2001) and Polonium Program.
References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10]
A. Haury, et al., Phys. Rev. Lett. 79 (1997) 511. H. Boukari, et al., Phys. Rev. Lett. 88 (2002) 207204. P. Kossacki, et al., Physica E 6 (2000) 709. P. Kossacki, et al., Physica E 12 (2002) 344. M. Goryca, D. Ferrand, P. Kossacki, M. Nawrocki, W. Pacuski, W. Mas´ lana, S. Tatarenko, J. Cibert, Phys. Stat. Sol. (b) (2006), to be published. D. Scalbert, et al., Solid State Comm. 66 (1988) 571. H.A. Sayad, S.M. Bhagat, Phys. Rev. B 31 (1985) 591. R.D. Kirby, et al., Phys. Rev. B 49 (1994) 10810. U. Nowak, et al., Phys. Rev. B 56 (1997) 8143. D. Kechrakos, et al., Phys. Rev. Lett. 94 (2005) 127201.