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Mechanism of the semiconductor-superconductor transition in amorphous chalcogenides
Journal of Non-Crystalline Solids 164-166 (1993) 1173-1176 North-Holland
MECHANISM TRANSITION Semen
D.
/OU~AL or ~I~LI,~ ~llI~
OF THE SEMICONDUCTOR-SUPERCONDUCTOR IN AMORPHOUS CHALCOGENIDES SA~SKY
Departamento de Fisica de Materiales, Facultad de Quimica, Universidad del Pals Vasco UPV/EHU, Ap4o 1072, San Sebastian 20080, Spain.
The experimental and theoretical studies of anomalous behaviour of the semiconductor - superconductor transition in chalcogenide glasses (ChG) are reviewed briefly. The superconducting characteristics of ChG correspond to the bipolaronic mechanism. The analysis of the superconductivity gives a new key for understanding the characteristics of local pairs in ChG. The local pairs c o n c e p t [i] successfully describes [2-5] various p r o p e r t i e s of chalcogenide glasses (ChG) having the n e g a t i v e effective Hubbard c o r r e l a t i o n energy U. About i0 years ago I predicted the p o s s i b i l i t y of bipolaronic superconductivity in the ChG m o d i f i e d by some dmetals due to a shift of the Fermi level to the m o b i l i t y edge [6]. At the same time two M o s c o w groups began to study superconductivity in ChG at high p r e s s u r e s [7-11]. Unusual s u p e r c o n d u c t i n g properties have been o b s e r v e d in ChG at p r e s s u r e s P about the dielectric-metal transition pressure P~ [7-10]. The superconductlvity transition temperature Tc increases rapidly at p r e s s u r e s P < Pt and more slowly when P > Pt" In a m o r p h o u s A g l 5 S i l 5 T e 7 0 and Se the d e p e n d e n c e Tc(P ) has a m a x i m u m at P = Pt [8,11]. The w i d t h of the s u p e r c o n d u c t i n g transition 6T c is considerable w h i c h t e s t i f i e s to great f l u c t u a t i o n s near Tc, but the 6Tc/T c d e r i v a t i v e de~22-3~3~3~06.~
creases as P grows [7-10]. The upper critical magnetic field Hc2 at any t e m p e r a t u r e T below T c as well as the conductivity above Tc increase as P grows. At the same P the values of T c and HG2 in amorphous alloys are h l g h e r than in the c r y s t a l l i ne samples with same chemical composition [9-11]. The dep e n d e n c e of Hc~ on t e m p e r a t u re T has a p o s l t i v e c u r v a t u r e in Agl5Sil5Te70 and in As2Te 3 the c u r v a t u r e changes from negative to p o s i t i v e as P increases. The critical c u r r e n t d e n s i t y J in As2Te 3 increases c o n s i d e r a b l y at Pt" The lower critical magnetic field Hcl is smaller than the Earth m a g n e t i c field. Atomic s t r u c t u r e of g l a s s y As2Te 3 is stable at high p r e s s u r e s P l'l*Pt" Positive values of the d i f f e r e n c e Z b e t w e e n T c values on glassy and c r y s t a l l i ne alloys as well as the positive d e r i v a t i v e of d Z / d P at any p r e s s u r e s smaller than Pt d i s a g r e e [12-14] with the theory of the influence of
© 1993 - Elsevier Sci~ce ~ b l i s h ~ s B.V. All rights reserve.
1174
S.D. Savransky / Semiconductor-superconductor transition in amorphous chalcogenides
A n d e r s o n l o c a l i z a t i o n on superconductivity [15] as well as the theory of T~ enhancement due to the interactlon between electrons and soft atomic p o t e n t i a l s [3-5] within the framework of the Bardeen-Cooper-Schrieffer (BCS) mechanism of superconductivity [16]. All peculiarities of the semiconductor - superconductor t r a n s i t i o n in ChG are in q u a l i t a t i v e a g r e e m e n t [17-19] with the local pair (bipolarons) theory of s u p e r c o n d u c tivity [20-22]. A supercond u c t i n g t r a n s i t i o n is possible in the c h a r g e d r e p u l s i v e boson system [20,23]. The local pairs are c r e a t e d in d i s o r d e r e d regions of ChG due to the self-trapping at intrinsic soft atomic potentials with small elastic constants k, which play an important role in the lowt e m p e r a t u r e p r o p e r t i e s of ChG [3-5]. The g r o w i n g pressure P destroys the intrinsic soft atomic potentials with increasing elastic constants, which influences on the local pairs' concentration. The average value of the polaronic shift ED very s t r o n g l y depends on ~ and E D decreases as P grows. At high P, when the Fermi level practically coincides with the m o b i l i t y edge, these iocal pairs c o u l d move as bipolarons. At low T these bipolarons could condense (because they are bosons) leading to s u p e r c o n d u c t i v i t y at Tc = n 2 / 3 / m *, where m~ is the b i p o l a r o n mass and n is their concentration in the
superconducting state. The coherence length Y in this theory obtained from the relation ¥ = (0/2WHc2)I/2, where # is the flux quantum. The local pair c o n c e n t r a t i o n is given by n = R ( 3 / 2 ) ¥ -3, where R is the R i e m a n n Zeta function. Since the coherence length corresponds to the thermal de Broglie wavelength of local pairs as ¥ = it gives an estimate for m* in the unit of the e l e c t r o n mass [19]. It is r e a s o n a b l e to assume that the Frohlich and deformation i n t e r a c t i o n s of bipolarons with a lattice v i b r a t i o n mode with frequency f are r e a l i z e d in ChG. The actual frequenc~ is determined as f ~ R y m /¥~, where Ry is the Rydberg constant. Hence, the polaronic shift Ep = fG 2 of the initial electronic b a n d w i d t h D = fL I/2 leads to the b i p o l a r o n i c band with width t = D/L, where L and G are dimensionless constants of the electronelectron attraction and the electron - phonon coupling. The e s t i m a t e d value of the magnetic field penetration depth Q = (m*/16f#e2n)l/2 is very great (about i0 -u cm), where # is the absolute magnetic permittivity). The lower critical m a g n e t i c field Hcl = Y H c 2 1 n ( Q / Y ) / 2 Q can be comparable with the Earth's m a g n e t i c field. Numerical analysis of the transitions shows that practically all parameters (except Y ~ 30A) of SC in the ChG d e c r e a s e with increasing P. The e s t i m a t e s determined
(2~/m*Tc)I/2,
$ 2~. Savranslcy / Semiconductor-superconductor transition in amorphous chalcogenides
for the characteristics of ChG o b t a i n e d from the data [7-11] at v a r i o u s pressures are p r e s e n t e d in Ref. [19,24] and all these estimates (see the table i) c o r r e s p o n d to the q u a n t i t a t i v e predictions of the bipolaron theory of superconductivity, In c o m p a r i s o n with the conventional superconductors [23], in ChG the c o n c e n t r a t i ons of charge carriers are very low and their m a s s e s are sufficiently large and they move in the n a r r o w band t. Since the t r a n s f e r integral t of b i p o l a r o n s d e p e n d s exponentially on the distance between the b i p o l a r o n sites the a n o m a l o u s l y large d T c / d P values are o b s e r v e d [7-11]. Note that in e x t r e m e l y n a r r o w band the influence of the d i s o r d e r e d g l a s s y n e t w o r k on t is comparable with the i n f l u e n c e of the phonon and p o l a r o n i c d i s t o r t i o n s of the c r y s t a l l i n e lattice on the t value. In ChG IUI d e c r e a s e s and the interaction between bipolarons increases with P and they p r a c t i c a l l y compensate each other. The suppression of local pair's mass plays the major role in Tc growth under P. In the region where Tc d e c r e a s e s as P increases the effect of the r e d u c t i o n of the local pairs' c o n c e n t r a t i o n n d o m i n a t e s and a t r a n s i t i o n from b i p o l a r o n i c to the usual BCS m e c h a n i s m is possible, e.g. in Se [24]. It is well known that impurities do not change the various properties of ChG [2]. But b i p o l a r o n s can be scattered by the intrinsic
1175
soft potentials, which the m e d i u m - r a n g e order creates in ChG [25], b e c a u s e the latter have sizes about a local pair's size at high P. Therefore, in ChG the "intrinsic dirty" case of Hc2 behaviour is realized [24] and H c 2 = A d * [ ( I - $ 3 / 2 ) / S ] 3 J 2, where S = T/T c [21], A d is proportional to the square root of the b i p o l a r o n s concentration. Since wide critical regime is r e a l i z e d in ChG. the n changes as (l-S) 2/3. its c o e f f i c i e n t is Ad-(I-S)I/3 as is c o n f i r m e d by recent analyses of the experiments [24]. The i m p o s s i b i l i t y of the 'clear" limit in any ChG is c o n f i r m e d by the i n d e p e n d e n c e of the A d c o e f f i c i e n t s on P. Table 1 M e d i u m v a l u e s of p a r a m e t e r s Chemical compositions
P a r a m e t e r s at OK n*1018 m* f (cm -3) /me/ (THz)
Ge-As-Se Ge2Se 3 GeTe As2Te 3 Si-Ag-Te
36 15 44 4.5 8
45 I0 31 25 220
1 2.5 9.1 12.6 1.4
Therefore, b i p o l a r o n i c superconductivity is r e a l i z e d in ChG, but further studies are necessary to clarify the detailed mechanism of the semiconductor - superconductor t r a n s i t i o n in ChG. From a p r a c t i c a l point of view, I stress again that the conditions for the semiconductor-superconductor transi-
1176
S J). Savransky / Semiconductor-superconductor transition in amorphous chalcogenides
tion could be softened in ChG m o d i f i e d by d-metals and Tc could increase under influence of some fields [26,18]. It is possible that ChG with a simple m a g n e t i c subsystem will be c o n v e n i e n t m a t e r i a l s for investigation of the bipolaronic superconductivity in d i s o r d e r e d systems. References i. P.W. Anderson, Phys. Rev. Lett., 34 (1975) 953. 2. N.F Mott and E.A. Davis, Electronic Processes in NonCrystalline Materials, C l a r e n d o n Press, Oxford 1979. 3. M.I. Klinger, Phys. Repts., 165 (1988) 275. 4. Yu.M. Galperin, V.G. K a r p o v and V.I. Kozub, Adv. Phys., 38 (1989) 669. 5. S.D. Savransky, Processes in Chalcogenide Glasses and Their Melts Conditioned by the Centres with Negative Correlation Energy. PhD Dissertation. (Leningrad, 1989) 165 pp. 6. S.D. Savransky, Proc 23 USSR Conf. "Low T e m p e r a t u r e Physics" Tallinn. 1 (1984) 218. 7. I.V. Berman, N.B. Brandt, V.A. Alekseev, I.E. Kostyleva, V.I. Sidorov and O.P. Pyatkina, JETF Lett., 40 (1984) 472. 8. A.A. Andreev, Berman I.V., K y s t a u b a e v T.Z., Meleh B.T., Sidorov V.I., Han Cuj-In. Fiz. Tver. Tela (Soviet Physics Solid State), 30 (1988) 2177. 9. V.A. Alekseev, I.V. Berman and V.A. Sidorov, Zeitsc. fur Phys. Chem., 156 (1988) 271.
i0. I.V. Berman and N.B. Brandt, Soviet Physics - J. Low Temperature Phys., 16 (1990) 1227. ii. M.A. Ii'ina, Fiz. Tver. Tela (Soviet Physics Solid State), 26 (1984) 5747. 12. S.D. Savransky, Fizika i Chimiya Stekla (Sov. J. Phys. Chem. Glasses) 16 (1990) 688. 13. S.D. Savransky, Materials science and Engineering, A145 (1991) 109. 14. S.D. Savransky, Solid State Communications, 79 (1991) 843. 15. G. Kotliar and A. Kapitulnik, Phys. Rev., B33 (1986) 3146. 16. N.M. Plakida, Physica Scripta, T29 (1989) 77. 17. S.D. Savransky, J. Non C r y s t a l . S o l i d s 83 (1986) 235. 18. S.D. Savransky, J. NonC r y s t a l . S o l i d s 94 (1987) 412. 19. S.D. Savransky, Mater. Res. Bull., 28 (1993) 81. 20. R. Micnas, J. R a n n i n g e r and S. Robaszkiewicz, Rev. Mod. Phys., 62 (1990) 113. 21. A.S. A l e x a n d r o v and A.B. Krebs, Uspechi Fizicheckih Nauk, 162 (1992) N5, p.l. 22. N.F. Mott, J. Phys. Condens. Matter., 5 (1993) 3487. 23. A.A. Abrikosov, Fundamentals of the Theory of Metals. Amsterdam, et.al., North-Holland, 1988. 614 pp. 24. S.D. Savransky, In preparation. 25. S.D. Savransky, Fizika i Chimiya Stekla (Sov. J. Phys. Chem. Glasses), 18 (1992) N6 p.135. 26. S.D. Savransky, Proc. 25 USSR Conf. on Low T e m p e r a t u r e Physics., Leningrad, 1 (1988) 34.
MECHANISM TRANSITION Semen
D.
/OU~AL or ~I~LI,~ ~llI~
OF THE SEMICONDUCTOR-SUPERCONDUCTOR IN AMORPHOUS CHALCOGENIDES SA~SKY
Departamento de Fisica de Materiales, Facultad de Quimica, Universidad del Pals Vasco UPV/EHU, Ap4o 1072, San Sebastian 20080, Spain.
The experimental and theoretical studies of anomalous behaviour of the semiconductor - superconductor transition in chalcogenide glasses (ChG) are reviewed briefly. The superconducting characteristics of ChG correspond to the bipolaronic mechanism. The analysis of the superconductivity gives a new key for understanding the characteristics of local pairs in ChG. The local pairs c o n c e p t [i] successfully describes [2-5] various p r o p e r t i e s of chalcogenide glasses (ChG) having the n e g a t i v e effective Hubbard c o r r e l a t i o n energy U. About i0 years ago I predicted the p o s s i b i l i t y of bipolaronic superconductivity in the ChG m o d i f i e d by some dmetals due to a shift of the Fermi level to the m o b i l i t y edge [6]. At the same time two M o s c o w groups began to study superconductivity in ChG at high p r e s s u r e s [7-11]. Unusual s u p e r c o n d u c t i n g properties have been o b s e r v e d in ChG at p r e s s u r e s P about the dielectric-metal transition pressure P~ [7-10]. The superconductlvity transition temperature Tc increases rapidly at p r e s s u r e s P < Pt and more slowly when P > Pt" In a m o r p h o u s A g l 5 S i l 5 T e 7 0 and Se the d e p e n d e n c e Tc(P ) has a m a x i m u m at P = Pt [8,11]. The w i d t h of the s u p e r c o n d u c t i n g transition 6T c is considerable w h i c h t e s t i f i e s to great f l u c t u a t i o n s near Tc, but the 6Tc/T c d e r i v a t i v e de~22-3~3~3~06.~
creases as P grows [7-10]. The upper critical magnetic field Hc2 at any t e m p e r a t u r e T below T c as well as the conductivity above Tc increase as P grows. At the same P the values of T c and HG2 in amorphous alloys are h l g h e r than in the c r y s t a l l i ne samples with same chemical composition [9-11]. The dep e n d e n c e of Hc~ on t e m p e r a t u re T has a p o s l t i v e c u r v a t u r e in Agl5Sil5Te70 and in As2Te 3 the c u r v a t u r e changes from negative to p o s i t i v e as P increases. The critical c u r r e n t d e n s i t y J in As2Te 3 increases c o n s i d e r a b l y at Pt" The lower critical magnetic field Hcl is smaller than the Earth m a g n e t i c field. Atomic s t r u c t u r e of g l a s s y As2Te 3 is stable at high p r e s s u r e s P l'l*Pt" Positive values of the d i f f e r e n c e Z b e t w e e n T c values on glassy and c r y s t a l l i ne alloys as well as the positive d e r i v a t i v e of d Z / d P at any p r e s s u r e s smaller than Pt d i s a g r e e [12-14] with the theory of the influence of
© 1993 - Elsevier Sci~ce ~ b l i s h ~ s B.V. All rights reserve.
1174
S.D. Savransky / Semiconductor-superconductor transition in amorphous chalcogenides
A n d e r s o n l o c a l i z a t i o n on superconductivity [15] as well as the theory of T~ enhancement due to the interactlon between electrons and soft atomic p o t e n t i a l s [3-5] within the framework of the Bardeen-Cooper-Schrieffer (BCS) mechanism of superconductivity [16]. All peculiarities of the semiconductor - superconductor t r a n s i t i o n in ChG are in q u a l i t a t i v e a g r e e m e n t [17-19] with the local pair (bipolarons) theory of s u p e r c o n d u c tivity [20-22]. A supercond u c t i n g t r a n s i t i o n is possible in the c h a r g e d r e p u l s i v e boson system [20,23]. The local pairs are c r e a t e d in d i s o r d e r e d regions of ChG due to the self-trapping at intrinsic soft atomic potentials with small elastic constants k, which play an important role in the lowt e m p e r a t u r e p r o p e r t i e s of ChG [3-5]. The g r o w i n g pressure P destroys the intrinsic soft atomic potentials with increasing elastic constants, which influences on the local pairs' concentration. The average value of the polaronic shift ED very s t r o n g l y depends on ~ and E D decreases as P grows. At high P, when the Fermi level practically coincides with the m o b i l i t y edge, these iocal pairs c o u l d move as bipolarons. At low T these bipolarons could condense (because they are bosons) leading to s u p e r c o n d u c t i v i t y at Tc = n 2 / 3 / m *, where m~ is the b i p o l a r o n mass and n is their concentration in the
superconducting state. The coherence length Y in this theory obtained from the relation ¥ = (0/2WHc2)I/2, where # is the flux quantum. The local pair c o n c e n t r a t i o n is given by n = R ( 3 / 2 ) ¥ -3, where R is the R i e m a n n Zeta function. Since the coherence length corresponds to the thermal de Broglie wavelength of local pairs as ¥ = it gives an estimate for m* in the unit of the e l e c t r o n mass [19]. It is r e a s o n a b l e to assume that the Frohlich and deformation i n t e r a c t i o n s of bipolarons with a lattice v i b r a t i o n mode with frequency f are r e a l i z e d in ChG. The actual frequenc~ is determined as f ~ R y m /¥~, where Ry is the Rydberg constant. Hence, the polaronic shift Ep = fG 2 of the initial electronic b a n d w i d t h D = fL I/2 leads to the b i p o l a r o n i c band with width t = D/L, where L and G are dimensionless constants of the electronelectron attraction and the electron - phonon coupling. The e s t i m a t e d value of the magnetic field penetration depth Q = (m*/16f#e2n)l/2 is very great (about i0 -u cm), where # is the absolute magnetic permittivity). The lower critical m a g n e t i c field Hcl = Y H c 2 1 n ( Q / Y ) / 2 Q can be comparable with the Earth's m a g n e t i c field. Numerical analysis of the transitions shows that practically all parameters (except Y ~ 30A) of SC in the ChG d e c r e a s e with increasing P. The e s t i m a t e s determined
(2~/m*Tc)I/2,
$ 2~. Savranslcy / Semiconductor-superconductor transition in amorphous chalcogenides
for the characteristics of ChG o b t a i n e d from the data [7-11] at v a r i o u s pressures are p r e s e n t e d in Ref. [19,24] and all these estimates (see the table i) c o r r e s p o n d to the q u a n t i t a t i v e predictions of the bipolaron theory of superconductivity, In c o m p a r i s o n with the conventional superconductors [23], in ChG the c o n c e n t r a t i ons of charge carriers are very low and their m a s s e s are sufficiently large and they move in the n a r r o w band t. Since the t r a n s f e r integral t of b i p o l a r o n s d e p e n d s exponentially on the distance between the b i p o l a r o n sites the a n o m a l o u s l y large d T c / d P values are o b s e r v e d [7-11]. Note that in e x t r e m e l y n a r r o w band the influence of the d i s o r d e r e d g l a s s y n e t w o r k on t is comparable with the i n f l u e n c e of the phonon and p o l a r o n i c d i s t o r t i o n s of the c r y s t a l l i n e lattice on the t value. In ChG IUI d e c r e a s e s and the interaction between bipolarons increases with P and they p r a c t i c a l l y compensate each other. The suppression of local pair's mass plays the major role in Tc growth under P. In the region where Tc d e c r e a s e s as P increases the effect of the r e d u c t i o n of the local pairs' c o n c e n t r a t i o n n d o m i n a t e s and a t r a n s i t i o n from b i p o l a r o n i c to the usual BCS m e c h a n i s m is possible, e.g. in Se [24]. It is well known that impurities do not change the various properties of ChG [2]. But b i p o l a r o n s can be scattered by the intrinsic
1175
soft potentials, which the m e d i u m - r a n g e order creates in ChG [25], b e c a u s e the latter have sizes about a local pair's size at high P. Therefore, in ChG the "intrinsic dirty" case of Hc2 behaviour is realized [24] and H c 2 = A d * [ ( I - $ 3 / 2 ) / S ] 3 J 2, where S = T/T c [21], A d is proportional to the square root of the b i p o l a r o n s concentration. Since wide critical regime is r e a l i z e d in ChG. the n changes as (l-S) 2/3. its c o e f f i c i e n t is Ad-(I-S)I/3 as is c o n f i r m e d by recent analyses of the experiments [24]. The i m p o s s i b i l i t y of the 'clear" limit in any ChG is c o n f i r m e d by the i n d e p e n d e n c e of the A d c o e f f i c i e n t s on P. Table 1 M e d i u m v a l u e s of p a r a m e t e r s Chemical compositions
P a r a m e t e r s at OK n*1018 m* f (cm -3) /me/ (THz)
Ge-As-Se Ge2Se 3 GeTe As2Te 3 Si-Ag-Te
36 15 44 4.5 8
45 I0 31 25 220
1 2.5 9.1 12.6 1.4
Therefore, b i p o l a r o n i c superconductivity is r e a l i z e d in ChG, but further studies are necessary to clarify the detailed mechanism of the semiconductor - superconductor t r a n s i t i o n in ChG. From a p r a c t i c a l point of view, I stress again that the conditions for the semiconductor-superconductor transi-
1176
S J). Savransky / Semiconductor-superconductor transition in amorphous chalcogenides
tion could be softened in ChG m o d i f i e d by d-metals and Tc could increase under influence of some fields [26,18]. It is possible that ChG with a simple m a g n e t i c subsystem will be c o n v e n i e n t m a t e r i a l s for investigation of the bipolaronic superconductivity in d i s o r d e r e d systems. References i. P.W. Anderson, Phys. Rev. Lett., 34 (1975) 953. 2. N.F Mott and E.A. Davis, Electronic Processes in NonCrystalline Materials, C l a r e n d o n Press, Oxford 1979. 3. M.I. Klinger, Phys. Repts., 165 (1988) 275. 4. Yu.M. Galperin, V.G. K a r p o v and V.I. Kozub, Adv. Phys., 38 (1989) 669. 5. S.D. Savransky, Processes in Chalcogenide Glasses and Their Melts Conditioned by the Centres with Negative Correlation Energy. PhD Dissertation. (Leningrad, 1989) 165 pp. 6. S.D. Savransky, Proc 23 USSR Conf. "Low T e m p e r a t u r e Physics" Tallinn. 1 (1984) 218. 7. I.V. Berman, N.B. Brandt, V.A. Alekseev, I.E. Kostyleva, V.I. Sidorov and O.P. Pyatkina, JETF Lett., 40 (1984) 472. 8. A.A. Andreev, Berman I.V., K y s t a u b a e v T.Z., Meleh B.T., Sidorov V.I., Han Cuj-In. Fiz. Tver. Tela (Soviet Physics Solid State), 30 (1988) 2177. 9. V.A. Alekseev, I.V. Berman and V.A. Sidorov, Zeitsc. fur Phys. Chem., 156 (1988) 271.
i0. I.V. Berman and N.B. Brandt, Soviet Physics - J. Low Temperature Phys., 16 (1990) 1227. ii. M.A. Ii'ina, Fiz. Tver. Tela (Soviet Physics Solid State), 26 (1984) 5747. 12. S.D. Savransky, Fizika i Chimiya Stekla (Sov. J. Phys. Chem. Glasses) 16 (1990) 688. 13. S.D. Savransky, Materials science and Engineering, A145 (1991) 109. 14. S.D. Savransky, Solid State Communications, 79 (1991) 843. 15. G. Kotliar and A. Kapitulnik, Phys. Rev., B33 (1986) 3146. 16. N.M. Plakida, Physica Scripta, T29 (1989) 77. 17. S.D. Savransky, J. Non C r y s t a l . S o l i d s 83 (1986) 235. 18. S.D. Savransky, J. NonC r y s t a l . S o l i d s 94 (1987) 412. 19. S.D. Savransky, Mater. Res. Bull., 28 (1993) 81. 20. R. Micnas, J. R a n n i n g e r and S. Robaszkiewicz, Rev. Mod. Phys., 62 (1990) 113. 21. A.S. A l e x a n d r o v and A.B. Krebs, Uspechi Fizicheckih Nauk, 162 (1992) N5, p.l. 22. N.F. Mott, J. Phys. Condens. Matter., 5 (1993) 3487. 23. A.A. Abrikosov, Fundamentals of the Theory of Metals. Amsterdam, et.al., North-Holland, 1988. 614 pp. 24. S.D. Savransky, In preparation. 25. S.D. Savransky, Fizika i Chimiya Stekla (Sov. J. Phys. Chem. Glasses), 18 (1992) N6 p.135. 26. S.D. Savransky, Proc. 25 USSR Conf. on Low T e m p e r a t u r e Physics., Leningrad, 1 (1988) 34.