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Luminescence of free and self-trapped excitons in wide-gap oxides
Journal of Luminescence 87}89 (2000) 232}234
Luminescence of free and self-trapped excitons in wide-gap oxides A. Lushchik *, M. Kirm , Ch. Lushchik , I. Martinson, G. Zimmerer Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia II Institute of Experimental Physics, University of Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany Department of Physics, Lund University, Professorsgatan 1, S 22100 Lund, Sweden
Abstract Behaviour of the intrinsic emissions of 7.6, 3.45 and 3.6 eV in Al O , Y O and Sc O , respectively, has been studied under a crystal excitation by 5}35 eV photons at 8}80 K. In Al O and Y O , electrons and holes do not undergo transformation into a self-trapped state, while self-shrunk excitons are formed at the direct optical creation of excitons or at the recombination of electrons with holes under the conditions of a high excitation density. In line with the theoretical assumption made by Sumi, manifestations of the photocreation of spatially correlated self-trapped d-electrons and p-holes are revealed in the spectra of the tunnel phosphorescence of Sc O . 2000 Elsevier Science B.V. All rights reserved. Keywords: Excitons; Self-trapping
Two types of excitons were revealed in solids long ago. Free excitons (FE) have been studied in detail in a number of semiconductors, while self-trapped excitons (STE) have been detected in wide-gap crystals. FE manifest themselves as narrow lines at the long-wavelength edge of a fundamental absorption. Broad bands of intrinsic emission are connected with relaxed STE [1]. In some alkali iodides, FE and STE coexist and their energy states are separated by an activation barrier [1,2]. A long-term detailed study of metal halides led to a conclusion that the existence of STE correlates with the self-trapping of p holes (V centres in alkali halides) or self-trapping of ) d electrons (lead halides), while s electrons or holes do not undergo transformation into a self-trapped state [2]. The creation of STE takes place at the direct photocreation of anion excitons as well as at the recombination of conduction electrons (e) with self-trapped holes (h ) or Q free holes (h) with self-trapped electrons (e ). Q * Corresponding author. Tel.: #372-7-428-946; fax: #3727-383-033. E-mail address: luch@".tartu.ee (A. Lushchik)
The behaviour of excitons in wide-gap oxides (WGO) is more complicated and is of great variety. In a MgO cubic crystal with the energy gap of E "7.85 eV, the emission of FE was detected long ago, while neither the luminescence of STE nor e and h has been observed as Q Q yet (see Ref. [3] and references therein). On the other hand, FE and STE coexist in a BeO crystal (E "10.3 eV) with a lower symmetry [4]. In Al O and Y O , the line-edge emission of excitons is not found, but a broadband intrinsic luminescence peaked at 7.6 and 3.45 eV can be excited in the region of weak longwavelength bands of a fundamental absorption of Al O (9 eV) and Y O (6 eV), respectively [5,6]. The quantum e$ciency of these emissions is g*0.3 at 5 K. In contrast to STE emission in alkali halides, the 7.6 eV emission of Al O and the 3.45 eV emission of Y O were not detec ted in the photo- and thermally stimulated luminescence of preliminary-irradiated crystals. On the basis of the phase diagram for a possible interaction of intrinsic electronic excitations with the "eld of acoustic phonons [7] it was supposed that in spite of the absence of e and h , the STE can be formed Q Q in Y O and Al O [8]. The sum of deformation
0022-2313/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 2 7 1 - 9
A. Lushchik et al. / Journal of Luminescence 87}89 (2000) 232}234
potentials of an electron and a hole is su$cient for the recombination formation of a localised exciton. A detailed study of a-Al O crystals of high purity (grown by the Czochralski technique) using synchrotron spectroscopy con"rmed the existence of such immobile selfshrunk excitons (SSE) even if an electron and a hole do not separately undergo the self-trapping [9]. In the present study, we continued the investigation of Al O crystals and compared the peculiarities of the excitation of 7.6 eV emission in Al O with the behav iour of 3.45 eV intrinsic luminescence in Y O . The exci tation spectra were measured for the time-integrated luminescence of WGO, as well as for the emission detected within a time window (length *t) correlated with the excitation pulses of synchrotron radiation (SR) delayed by dt. The photoluminescence experiments with time resolution were carried out at the SUPERLUMI station of HASYLAB at DESY, Hamburg. Some of the SR measurements were performed at beamline 52 in the MAX-I Laboratory in Lund, Sweden. Fig. 1 presents the excitation spectra for the time-integrated, fast (*t" 13.8 ns, dt"2.4 ns) and slow (*t"110 ns, dt"56 ns) components of 7.6 eV emission (recorded through a vacuum monochromator) in a Al O crystal at 8 K. An inset shows the emission spectrum of a-Al O at 8 K. A fast component of 7.6 eV luminescence dominates in the emission spectrum of Al O at 8 K. This component can be e!ectively excited in the region of 8.85}9.15 eV and the excitation maximum is located at 8.97 eV. The e$ciency maximum for a slow component is shifted toward the higher energy. The excitation of these fast- and slowemission components corresponds to the formation of singlet and triplet excitons with p-type hole and s-type electron components. According to Fig. 1, the e$ciency of the 7.6 eV emission is extremely low in the region of hl *9.4 eV, where exciting photons generate e}h pairs. The e$ciency of the slow component of STE emission in the region of 24}36 eV is approximately the same as at the direct formation of excitons by 9.1 eV photons. Photons of 24}36 eV belong to the region of the multiplication of electronic excitations (MEE) in Al O , if one absorbed photon is able to create two or three e}h pairs [9]. Fig. 2 depicts the excitation spectrum of 3.45 eV emission and the emission spectrum of Y O at 8 K. The time-integrated component of an intrinsic emission is e$ciently excited at the direct formation of excitons by 6.0}6.5 eV photons. A slow component (*t"130 ns, dt"30 ns) dominates in the emission band of STE in Y O at 8 K. The e$ciency of exciton emission signi" cantly decreases at hl 'E 6.5 eV, while its sharp increase occurs in the region of 14.5}18.0 eV. It was shown earlier [10] that one absorbed photon of 15}18 eV causes the formation of both an e}h pair and a secondary exciton, while two e}h pairs can be created by one exciting photon of hl '18 eV. Y> ions are mainly respon
233
Fig. 1. Excitation spectra for the time-integrated (solid line), fast [*t"13.8 ns, dt"2.4 ns (dashed line)] and slow [*t"110 ns, dt"56 ns (*)] components of 7.6 eV emission in Al O at 8 K. Inset shows the main band of intrinsic emission in Al O at 8 K.
Fig. 2. Excitation spectrum for the time-integrated 3.45 eV emission in Y O at 8 K. Inset shows the emission spectrum of Y O at the excitation in the region of fundamental absorption at 8 K.
sible for the absorption of 29}34 eV exciting photons. The e$ciency of the recombination creation of triplet excitons at a high local density of e}h pairs in the MEE region (25}35 eV) signi"cantly exceeds the relevant value at the formation of single e}h pairs by 10}12 eV photons. In Al O and Y O at 8 K, the energy gap equals 9.4 and 6.3 eV, respectively. The comparison of the data presented in Figs. 1 and 2 allows us to conclude that the excitation spectra of the slow component of intrinsic exciton luminescence in Al O and slow exciton emis sion in Y O are similar (accurate to a scale factor of E (Al O )/E (Y O )), but they di!er signi"cantly from the excitation spectra of STE emission in alkali halides, where the e$ciency of exciton luminescence at the photocreation of single e}h pairs is approximately the same as at the direct formation of excitons. In Al O and Y O , the e$ciency of STE emission at the creation of single e}h pairs (10}23 eV in Al O and 9}14 eV in Y O ) by exciting photons (I +10 photon/s) is tens of times as low as at the direct optical formation of excitons. The dependence of the intensity of the 7.6 eV luminescence on the density of excitation by nanosecond electron pulses has been studied for a Al O crystal. It was shown that the e$ciency of STE emission is by more than an order of magnitude higher if a single electron pulse forms more than 10 e}h pairs in 1 cm of an excited crystal [9]. A similar e!ect has been detected in Y O as well. In the
234
A. Lushchik et al. / Journal of Luminescence 87}89 (2000) 232}234
case of SR the density of excitation is several orders of magnitude lower than 10 e}h/cm. Under the conditions of MEE, when one absorbed photon causes the creation of two or three e}h pairs, the local density of an excitation sharply increases in comparison with the excitation by photons forming single e}h pairs. According to Figs. 1 and 2, the e$ciency of STE emission signi"cantly increases in the region of MEE. An e!ective cross-section for the recombination of free e and h in a regular lattice is hundreds of times lower than the value for e}h recombination in case when one of the carriers is localized. Therefore, an e!ective recombination creation of STE in Al O and Y O occurs at a signi"cantly higher excita tion density than that in alkali halides, if the recombination formation of excitons takes place via a hole self-trapping (e recombines with h ). Q Besides SSE formed due to the addition of deformation potentials of a hole and an electron, the Sumi phase diagram [7] predicts another unusual situation: a photon in the region of band-to-band transitions creates a metastable exciton that decays with the formation of spatially separated e and h . The sequence of s and d conduction Q Q bands has been analysed for a Sc O crystal (isostruc tural to Y O ) [11]. It was shown that the bottom of the conduction band may be determined by d states. So, one might expect in Sc O the formation of an electronic excitation with a p-hole component and a d-electron component. The deformation potentials for these components are rather di!erent providing the instability of excitons and the formation of spatially correlated e and h . Q Q The irradiation of a Sc O single crystal by 6}36 eV photons or 2}10 keV electrons causes the appearance of an intense emission with the maximum of 3.6 eV (see inset in Fig. 3). Fig. 3 also shows the excitation spectrum of this emission (time-integrated) at 8 K. In contrast to Al O and Y O , we failed to detect even a weak re#ec tion maximum near the edge of a fundamental absorption, that can be ascribed to the formation of excitons. The e$ciency of 3.6 eV emission has a maximum at 6.2 eV and it decreases in the region of 8}14 eV. A sharp increase of 3.6 eV luminescence e$ciency occurs at hl "15}17 eV and in the region of 27}30 eV. The most fascinating di!erence between the behaviour of the intrinsic emission in Sc O and Al O or Y O lies in the fact that a slowly damping phosphorescence with the same emission spectrum (see inset in Fig. 3) as that under photoexcitation is observed for several minutes after the excitation of Sc O at ¹"8}70 K has been stopped. The decay kinetics of this phosphorescence does not practically depend on temperature. Such behaviour is
Fig. 3. Excitation spectrum of time-integrated 3.6 eV emission in Sc O at 8 K. The spectrum of intrinsic emission (solid line) and the phosphorescence spectrum (*) for a Sc O crystal at 8 K.
typical of a tunnel luminescence con"rming the presence of pairs of spatially correlated e and h in Sc O at Q Q 8}70 K. Certainly, a detailed further study of Sc O is needed.
Acknowledgements This work was supported by the Estonian Science Foundation (grant No. 3868), the Craaford and the STINT Foundation (Sweden) and the Exchange Program between Tartu and Hamburg Universities.
References [1] E.I. Rashba, M.D. Sturge (Eds.), Excitons, North-Holland, Amsterdam, 1982. [2] Ch. Lushchik, I. Kuusmann, V. Plekhanov, J. Lumin. 18/19 (1979) 11. [3] E. Feldbach, Ch. Lushchik, I. Kuusmann, Sov. JETP Lett. 39 (1984) 61. [4] V.Y. Ivanov, V.A. Pustovarov, S.V. Gorbunov, S.V. Kudyakov, A.V. Kruzhalov, Phys. Solid State 38 (1996) 1818. [5] A. Kuznetsov, B. Namozov, V. MuK rk, Proc. Acad. Sci. Estonian SSR, Phys. Math. 36 (1987) 193. [6] A.I. Kuznetsov, V.N. Abramov, N.S. Rooze, T.I. Savikhina, Sov. JETP Lett. 28 (1978) 602. [7] A. Sumi, J. Phys. Soc. Japan 43 (1977) 1286. [8] Ch.B. Lushchik, Izv. Akad. Nauk Latvian SSR, Ser. Fiz. Techn. Nauk N 2 (1984) 57. [9] M. Kirm, G. Zimmerer, E. Feldbach, A. Lushchik, Ch. Lushchik, F. Savikhin, Phys. Rev. B 60 (1999) 502. [10] Yu.M. Aleksandrov, Ch.B. Lushchik, V.N. Makhov, M.N. Yakimenko, Bull. Acad. Sci. USSR, Phys. Ser. 49 (1985) 160. [11] V.N. Abramov, A.N. Ermoshkin, A.I. Kuznetsov, V. MuK rk, Phys. Stat. Sol. B 121 (1984) K59.
Luminescence of free and self-trapped excitons in wide-gap oxides A. Lushchik *, M. Kirm , Ch. Lushchik , I. Martinson, G. Zimmerer Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia II Institute of Experimental Physics, University of Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany Department of Physics, Lund University, Professorsgatan 1, S 22100 Lund, Sweden
Abstract Behaviour of the intrinsic emissions of 7.6, 3.45 and 3.6 eV in Al O , Y O and Sc O , respectively, has been studied under a crystal excitation by 5}35 eV photons at 8}80 K. In Al O and Y O , electrons and holes do not undergo transformation into a self-trapped state, while self-shrunk excitons are formed at the direct optical creation of excitons or at the recombination of electrons with holes under the conditions of a high excitation density. In line with the theoretical assumption made by Sumi, manifestations of the photocreation of spatially correlated self-trapped d-electrons and p-holes are revealed in the spectra of the tunnel phosphorescence of Sc O . 2000 Elsevier Science B.V. All rights reserved. Keywords: Excitons; Self-trapping
Two types of excitons were revealed in solids long ago. Free excitons (FE) have been studied in detail in a number of semiconductors, while self-trapped excitons (STE) have been detected in wide-gap crystals. FE manifest themselves as narrow lines at the long-wavelength edge of a fundamental absorption. Broad bands of intrinsic emission are connected with relaxed STE [1]. In some alkali iodides, FE and STE coexist and their energy states are separated by an activation barrier [1,2]. A long-term detailed study of metal halides led to a conclusion that the existence of STE correlates with the self-trapping of p holes (V centres in alkali halides) or self-trapping of ) d electrons (lead halides), while s electrons or holes do not undergo transformation into a self-trapped state [2]. The creation of STE takes place at the direct photocreation of anion excitons as well as at the recombination of conduction electrons (e) with self-trapped holes (h ) or Q free holes (h) with self-trapped electrons (e ). Q * Corresponding author. Tel.: #372-7-428-946; fax: #3727-383-033. E-mail address: luch@".tartu.ee (A. Lushchik)
The behaviour of excitons in wide-gap oxides (WGO) is more complicated and is of great variety. In a MgO cubic crystal with the energy gap of E "7.85 eV, the emission of FE was detected long ago, while neither the luminescence of STE nor e and h has been observed as Q Q yet (see Ref. [3] and references therein). On the other hand, FE and STE coexist in a BeO crystal (E "10.3 eV) with a lower symmetry [4]. In Al O and Y O , the line-edge emission of excitons is not found, but a broadband intrinsic luminescence peaked at 7.6 and 3.45 eV can be excited in the region of weak longwavelength bands of a fundamental absorption of Al O (9 eV) and Y O (6 eV), respectively [5,6]. The quantum e$ciency of these emissions is g*0.3 at 5 K. In contrast to STE emission in alkali halides, the 7.6 eV emission of Al O and the 3.45 eV emission of Y O were not detec ted in the photo- and thermally stimulated luminescence of preliminary-irradiated crystals. On the basis of the phase diagram for a possible interaction of intrinsic electronic excitations with the "eld of acoustic phonons [7] it was supposed that in spite of the absence of e and h , the STE can be formed Q Q in Y O and Al O [8]. The sum of deformation
0022-2313/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 2 7 1 - 9
A. Lushchik et al. / Journal of Luminescence 87}89 (2000) 232}234
potentials of an electron and a hole is su$cient for the recombination formation of a localised exciton. A detailed study of a-Al O crystals of high purity (grown by the Czochralski technique) using synchrotron spectroscopy con"rmed the existence of such immobile selfshrunk excitons (SSE) even if an electron and a hole do not separately undergo the self-trapping [9]. In the present study, we continued the investigation of Al O crystals and compared the peculiarities of the excitation of 7.6 eV emission in Al O with the behav iour of 3.45 eV intrinsic luminescence in Y O . The exci tation spectra were measured for the time-integrated luminescence of WGO, as well as for the emission detected within a time window (length *t) correlated with the excitation pulses of synchrotron radiation (SR) delayed by dt. The photoluminescence experiments with time resolution were carried out at the SUPERLUMI station of HASYLAB at DESY, Hamburg. Some of the SR measurements were performed at beamline 52 in the MAX-I Laboratory in Lund, Sweden. Fig. 1 presents the excitation spectra for the time-integrated, fast (*t" 13.8 ns, dt"2.4 ns) and slow (*t"110 ns, dt"56 ns) components of 7.6 eV emission (recorded through a vacuum monochromator) in a Al O crystal at 8 K. An inset shows the emission spectrum of a-Al O at 8 K. A fast component of 7.6 eV luminescence dominates in the emission spectrum of Al O at 8 K. This component can be e!ectively excited in the region of 8.85}9.15 eV and the excitation maximum is located at 8.97 eV. The e$ciency maximum for a slow component is shifted toward the higher energy. The excitation of these fast- and slowemission components corresponds to the formation of singlet and triplet excitons with p-type hole and s-type electron components. According to Fig. 1, the e$ciency of the 7.6 eV emission is extremely low in the region of hl *9.4 eV, where exciting photons generate e}h pairs. The e$ciency of the slow component of STE emission in the region of 24}36 eV is approximately the same as at the direct formation of excitons by 9.1 eV photons. Photons of 24}36 eV belong to the region of the multiplication of electronic excitations (MEE) in Al O , if one absorbed photon is able to create two or three e}h pairs [9]. Fig. 2 depicts the excitation spectrum of 3.45 eV emission and the emission spectrum of Y O at 8 K. The time-integrated component of an intrinsic emission is e$ciently excited at the direct formation of excitons by 6.0}6.5 eV photons. A slow component (*t"130 ns, dt"30 ns) dominates in the emission band of STE in Y O at 8 K. The e$ciency of exciton emission signi" cantly decreases at hl 'E 6.5 eV, while its sharp increase occurs in the region of 14.5}18.0 eV. It was shown earlier [10] that one absorbed photon of 15}18 eV causes the formation of both an e}h pair and a secondary exciton, while two e}h pairs can be created by one exciting photon of hl '18 eV. Y> ions are mainly respon
233
Fig. 1. Excitation spectra for the time-integrated (solid line), fast [*t"13.8 ns, dt"2.4 ns (dashed line)] and slow [*t"110 ns, dt"56 ns (*)] components of 7.6 eV emission in Al O at 8 K. Inset shows the main band of intrinsic emission in Al O at 8 K.
Fig. 2. Excitation spectrum for the time-integrated 3.45 eV emission in Y O at 8 K. Inset shows the emission spectrum of Y O at the excitation in the region of fundamental absorption at 8 K.
sible for the absorption of 29}34 eV exciting photons. The e$ciency of the recombination creation of triplet excitons at a high local density of e}h pairs in the MEE region (25}35 eV) signi"cantly exceeds the relevant value at the formation of single e}h pairs by 10}12 eV photons. In Al O and Y O at 8 K, the energy gap equals 9.4 and 6.3 eV, respectively. The comparison of the data presented in Figs. 1 and 2 allows us to conclude that the excitation spectra of the slow component of intrinsic exciton luminescence in Al O and slow exciton emis sion in Y O are similar (accurate to a scale factor of E (Al O )/E (Y O )), but they di!er signi"cantly from the excitation spectra of STE emission in alkali halides, where the e$ciency of exciton luminescence at the photocreation of single e}h pairs is approximately the same as at the direct formation of excitons. In Al O and Y O , the e$ciency of STE emission at the creation of single e}h pairs (10}23 eV in Al O and 9}14 eV in Y O ) by exciting photons (I +10 photon/s) is tens of times as low as at the direct optical formation of excitons. The dependence of the intensity of the 7.6 eV luminescence on the density of excitation by nanosecond electron pulses has been studied for a Al O crystal. It was shown that the e$ciency of STE emission is by more than an order of magnitude higher if a single electron pulse forms more than 10 e}h pairs in 1 cm of an excited crystal [9]. A similar e!ect has been detected in Y O as well. In the
234
A. Lushchik et al. / Journal of Luminescence 87}89 (2000) 232}234
case of SR the density of excitation is several orders of magnitude lower than 10 e}h/cm. Under the conditions of MEE, when one absorbed photon causes the creation of two or three e}h pairs, the local density of an excitation sharply increases in comparison with the excitation by photons forming single e}h pairs. According to Figs. 1 and 2, the e$ciency of STE emission signi"cantly increases in the region of MEE. An e!ective cross-section for the recombination of free e and h in a regular lattice is hundreds of times lower than the value for e}h recombination in case when one of the carriers is localized. Therefore, an e!ective recombination creation of STE in Al O and Y O occurs at a signi"cantly higher excita tion density than that in alkali halides, if the recombination formation of excitons takes place via a hole self-trapping (e recombines with h ). Q Besides SSE formed due to the addition of deformation potentials of a hole and an electron, the Sumi phase diagram [7] predicts another unusual situation: a photon in the region of band-to-band transitions creates a metastable exciton that decays with the formation of spatially separated e and h . The sequence of s and d conduction Q Q bands has been analysed for a Sc O crystal (isostruc tural to Y O ) [11]. It was shown that the bottom of the conduction band may be determined by d states. So, one might expect in Sc O the formation of an electronic excitation with a p-hole component and a d-electron component. The deformation potentials for these components are rather di!erent providing the instability of excitons and the formation of spatially correlated e and h . Q Q The irradiation of a Sc O single crystal by 6}36 eV photons or 2}10 keV electrons causes the appearance of an intense emission with the maximum of 3.6 eV (see inset in Fig. 3). Fig. 3 also shows the excitation spectrum of this emission (time-integrated) at 8 K. In contrast to Al O and Y O , we failed to detect even a weak re#ec tion maximum near the edge of a fundamental absorption, that can be ascribed to the formation of excitons. The e$ciency of 3.6 eV emission has a maximum at 6.2 eV and it decreases in the region of 8}14 eV. A sharp increase of 3.6 eV luminescence e$ciency occurs at hl "15}17 eV and in the region of 27}30 eV. The most fascinating di!erence between the behaviour of the intrinsic emission in Sc O and Al O or Y O lies in the fact that a slowly damping phosphorescence with the same emission spectrum (see inset in Fig. 3) as that under photoexcitation is observed for several minutes after the excitation of Sc O at ¹"8}70 K has been stopped. The decay kinetics of this phosphorescence does not practically depend on temperature. Such behaviour is
Fig. 3. Excitation spectrum of time-integrated 3.6 eV emission in Sc O at 8 K. The spectrum of intrinsic emission (solid line) and the phosphorescence spectrum (*) for a Sc O crystal at 8 K.
typical of a tunnel luminescence con"rming the presence of pairs of spatially correlated e and h in Sc O at Q Q 8}70 K. Certainly, a detailed further study of Sc O is needed.
Acknowledgements This work was supported by the Estonian Science Foundation (grant No. 3868), the Craaford and the STINT Foundation (Sweden) and the Exchange Program between Tartu and Hamburg Universities.
References [1] E.I. Rashba, M.D. Sturge (Eds.), Excitons, North-Holland, Amsterdam, 1982. [2] Ch. Lushchik, I. Kuusmann, V. Plekhanov, J. Lumin. 18/19 (1979) 11. [3] E. Feldbach, Ch. Lushchik, I. Kuusmann, Sov. JETP Lett. 39 (1984) 61. [4] V.Y. Ivanov, V.A. Pustovarov, S.V. Gorbunov, S.V. Kudyakov, A.V. Kruzhalov, Phys. Solid State 38 (1996) 1818. [5] A. Kuznetsov, B. Namozov, V. MuK rk, Proc. Acad. Sci. Estonian SSR, Phys. Math. 36 (1987) 193. [6] A.I. Kuznetsov, V.N. Abramov, N.S. Rooze, T.I. Savikhina, Sov. JETP Lett. 28 (1978) 602. [7] A. Sumi, J. Phys. Soc. Japan 43 (1977) 1286. [8] Ch.B. Lushchik, Izv. Akad. Nauk Latvian SSR, Ser. Fiz. Techn. Nauk N 2 (1984) 57. [9] M. Kirm, G. Zimmerer, E. Feldbach, A. Lushchik, Ch. Lushchik, F. Savikhin, Phys. Rev. B 60 (1999) 502. [10] Yu.M. Aleksandrov, Ch.B. Lushchik, V.N. Makhov, M.N. Yakimenko, Bull. Acad. Sci. USSR, Phys. Ser. 49 (1985) 160. [11] V.N. Abramov, A.N. Ermoshkin, A.I. Kuznetsov, V. MuK rk, Phys. Stat. Sol. B 121 (1984) K59.