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Electronic conduction in (Bi,Pb)–Sr–Ca–Cu–O granular superconductors
Journal of Non-Crystalline Solids 354 (2008) 4323–4325
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Electronic conduction in (Bi,Pb)–Sr–Ca–Cu–O granular superconductors M. Gazda *, B. Kusz, L. Murawski ´ sk, Poland Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdan
a r t i c l e
i n f o
Article history: Available online 31 July 2008 PACS: 74.72.Hs 72.20.Ee 74.81Bd Keyword: Electrical and electronic properties
a b s t r a c t Granular superconductors are very interesting materials owing to their untypical electrical properties caused by the presence of Coulomb effects, electron tunnelling, Josephson coupling between granules and vario\us aspects of disorder. The electrical properties of (Bi,Pb)–Sr–Ca–Cu–O granular superconductors obtained by the solid state crystallization method have been studied in this paper. The materials may be considered as a system of granules of a high-temperature suprconductor embedded in the insulating matrix. The main parameter which determines the properties of granular materials is the tunnelling conductivity between the neighboring grains. Composed of small weakly coupled granules they are characterized by hopping conductivity exponential dependence. Samples containing relatively large granules of the 2212 phase (above 30 nm), with larger inter-grain conductivity, above the transition temperature, exhibit a logarithmic temperature dependence of resistivity. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction
rðTÞ ¼ r0 1 a ln
Granular materials are composed of conducting grains of a size between a few and a few hundred nanometers embedded in an insulating or semiconducting matrix. They have been studied for many years and several physical mechanisms of electronic transport phenomena have been proposed [e.g. 1–6]. The main physical parameter which determines the properties of granular materials is tunnelling conductivity between neighboring grains [7]. It is described as dimensionless conductance, g, expressed in units of quantum conductance e2 = h. When the inter-grain conductance is low (g < 1), the granular material is in a dielectric regime and its electrical conductivity occurs through thermally activated tunnelling of electrons between grains [1]. Moreover, Coulomb effects may influence the electronic transport. The temperature dependence of the resistivity of granular materials in the dielectric regime usually follows the exponential relation:
qðTÞ ¼ q0 e
T0 T
n
;
ð1Þ
where n is in the range 0.2 6 n 6 1 and depends on the geometry of conductivity and the details of the density of states of the granular material. In low temperatures n at a level close to 0.5 has been often observed [3]. When conductivity between the grains is larger (g > 1), but still smaller than inter-grain conductivity, the granular material is in the metallic regime and Coulomb interaction between the grains may be neglected. In this case, as shown by Efetov et al., conductivity follows a logarithmic relation [8]: * Corresponding author. Tel.: +48 58 347 14 86; fax: +48 58 347 28 21. E-mail address: [email protected] (M. Gazda). 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.06.044
gEc ; T
ð2Þ
where Ec is the electrostatic energy of adding one electron to a grain,
a ¼ ð2pdgÞ1
ð3Þ
depends on the tunnelling conductivity of grains, g, and the system dimensionality, d. Dimensionality d is considered as half of the grains coordination number. The relations (2) and Eq. (3) are valid if T >> gd, where d is the mean energy level spacing in a single grain [8]. It should be also emphasized that the inter-grain conductivity depends on the granular material phase composition and microstructure. The behavior of a system in which grains transit into a superconducting state is not fully understood. Experimental studies have given a variety of results. For instance, a very interesting behavior of a ‘superconductor–insulator’ was found by Gerber et al. [5]. They observed that the resistivity of a granular metal below the critical temperature was much higher than in the normal state. Chui et al. reported three types of the temperature dependence of granular aluminium resistivity: exponential, /T1/2 and logarithmic [6]. A transition to a global superconducting state has been observed both in the case of strong localization with exponential temperature dependence of resistivity and in a weak localization regime. The presented work concerns the electrical properties of (Bi,Pb)4Sr3Ca3Cu4Ox granular materials obtained by solid state crystallization. These materials may be considered as a 3D system of grains of a metallic phase embedded in an insulating matrix. The
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M. Gazda et al. / Journal of Non-Crystalline Solids 354 (2008) 4323–4325
insulating or semiconducting matrix in which metallic grains are located is composed both of amorphous material and crystalline non-metallic phases [9]. The metallic phases which are formed as a result of crystallization are oxide superconductors belonging to the bismuth family, i.e. (Bi,Pb)2Sr2CuOx (2201 with Tc = 10 K), and (Bi,Pb)2Sr2CaCu2Ox (2212, Tc = 85 K). Depending on the temperature and time of crystallization, the size and/or the number of conducting grains increases while the width of the insulating barriers between them decreases. Simultaneously, normal state intergrain tunnelling conductivity increases and superconducting coupling between the grains becomes stronger. As a result, the electrical properties of the material change. 2. Experimental The samples were produced by solid state crystallization. Firstly, the samples of (Bi0.8Pb0.2)4Sr3Ca3Cu4Ox glass were prepared from a reagent grade: Bi(NO3) 5H2O, PbO, CuO, Sr(NO3)2 and CaCO3. The substrates were mixed at the (Bi,Pb):Sr:Ca:Cu 4:3:3:4 ratio and calcinated at 820 °C for 10 h. Next, they were melted in a platinum crucible at 1250 °C, kept at a high temperature for about 10 min and quenched. The glass was cut into bars of similar dimensions (2 1 8 mm3) and polished before further thermal treatment. The crystallization was carried out in a tube furnace at temperatures between 750 and 850 °C. The samples were put into a furnace which was already hot and after they were quenched after appropriate time. The heat treatment conditions affect the amount of metallic phases and the granule radius. Generally speaking, the higher the temperature and the longer the annealing time is, the greater is the number of metallic phases distributed in the form of larger granules in the sample [9]. The samples were checked by X-ray diffraction (XRD), atomic force microscopy (AFM) and scanning electron microscopy (SEM) methods. Measurements of resistivity as a function of temperature were made by a DC technique in a standard four-terminal configuration at temperatures ranging between 3 and 300 K. 3. Results and discussion Granular materials obtained by solid state crystallization may by divided into three groups. One group consists of samples in which metallic granules remain in a normal state in the whole analyzed temperature range. The second group includes materials in which granules transit into a superconducting state below the critical temperature, however the coupling between them is too weak and no long-range superconducting state develops. In materials belonging to the third group the grains are significantly larger than in other groups and the coupling between them is strong enough to make the whole sample superconduct below the critical temperature. It is not possible to calculate the exact value of tunnelling conductance g, or to determine the electrical properties of the granular system in a real granular system. Following the example shown by Efetov et al. [8] we have attempted to make some very rough estimations based on the knowledge of room temperature resistivity and average grain dimensions. The results are shown in Table 1. An analysis of the temperature dependence of resistivity in a (Bi,Pb)SrCaCuO granular system of small granules (not larger than 25 nm) being in a normal state has been performed elsewhere [10]. In this case the tunnelling conductivity between granules is small (g 1) and the temperature dependence of resistivity follows the exponential low (Fig. 1). It has been concluded that electronic transport occurs by the variable range hopping either with or without Coulomb interactions. Coulomb interactions determine con-
Table 1 Microstructural and electrical properties of the studied samples: d is the average diameter of (Bi,Pb)2Sr2CuOx or (Bi,Pb)2Sr2CaCu2Oy metallic granules; q(300 K) is the resistivity at the room temperature, g is the inter-grain dimensionless conductance and a is a parameter describing the logarithmic temperature dependence of resistivity Sample description
q(300 K)
Average grain dimension (nm)
Grain shape
d
g
a
(X cm)
500 °C, 500 °C, 650 °C, 750 °C, 800 °C, 820 °C, 840 °C,
1h 2h 4 min 2 min 2 min 2 min 2 min
10 ± 1 3.4 ± 0.3 1.6 ± 0.1 21 ± 0.2 6.6 ± 0.2 3.8 ± 0.2 0.25 ± 0.05
8–10 21–25 10–27 15–20 30–40 30–40 30–40
Oval Oval Oval Oval Oval Oval Oval
3 3 3 3 3 3 3
Exponential q(T) relation
850 °C, 2 min
0.45 ± 0.05
30–40
Oval
3
750 °C, 750 °C, 800 °C, 850 °C, 820 °C,
0.09 ± 0.01 0.06 ± 0.01 0.05 ± 0.01 0.03 ± 0.01 0.01 ± 0.001
100 100 1000 5000 5000
Oval Oval Plate Plate Plate
3 3 2 2 2
102 0.1 0.1 0.1 1 1 1– 10 1– 10 >10 >10 >100 >100 >100
8 min 32 min 8 min 32 min 16 min
0.156 0.158 0.156 0.156 0.134 0.134 0.128 0.128 0.1
Fig. 1. Resistivity of samples annealed at temperatures ranging between 500 and 750 °C plotted as a function of Tn.
ductivity in case of material containing granules of about 10 nm [10]. Examples of the temperature dependence of resistivity in a (Bi,Pb)SrCaCuO granular system of larger granules (equal to or larger than 30 nm) in the temperature ranging between 100 and 300 K are shown in Fig. 2. The condition T dg is fulfilled well, since d is approx. 0.02 meV for a grain dimension of about 20 nm and it decreases rapidly with dimensions of grains [10]. Also all the samples have significantly larger inter-grain conductivities than the samples with exponential q(T) dependence (Table 1). However, it can be seen that plots of resistivity, not conductivity as a function of ln T, are linear at quite a wide temperature range. The temperature dependence of resistivity in these samples may be expressed by the following relation:
qðTÞ ¼ Að1 a ln TÞ:
ð4Þ
Eqs. (4) and (2) are not identical, however at some temperature ranges they express similar dependencies. Therefore, it may be assumed that parameter a in (4) depends on the same properties of the granular system as the parameter in (2). Parameters describing the logarithmic temperature dependence of resistivity are collected in Table 1. It can be seen that parameter a tends to decrease with the increase of g, in qualitative agreement with the relation
M. Gazda et al. / Journal of Non-Crystalline Solids 354 (2008) 4323–4325
4325
4. Conclusions The studied materials were 3D systems of grains of (Bi,Pb)2Sr2CuOx and (Bi,Pb)2Sr2CaCu2Ox phases embedded in an insulating matrix. The electrical properties of (Bi,Pb)4Sr3Ca3Cu4Ox granular materials have been studied and discussed within the model of Efetov et al. The resistivity of the samples containing small, weakly coupled grains in a normal state follows the exponential dependence on temperature q(T) / exp[(T/T0)n]. Granular samples with larger inter-grain tunnelling conductivities exhibit a logarithmic temperature dependence of resistivity. Parameter a describing q(T) is in a qualitative agreement with the theoretical predictions. Fig. 2. Temperature dependence of resistivity of samples annealed at temperatures ranging between 750 and 850 °C.
Eq. (3). In order to quantitatively test the validity of relations (2), (4), a more precise way of determining the tunnelling conductance would be necessary. It is not clear why relation (4) is fulfilled better than relation (2) in a (Bi,Pb)SrCaCuO granular system. The reason may be connected with the inevitable distribution of grain dimensions and shapes. It should be also noted that such a logarithmic temperature dependence of resistivity has been recently observed in several granular materials [e.g. 5].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
B. Abeles, P. Sheng, M.D. Coutts, Y. Arie, Adv. Phys. 24 (1975) 407. P. Sheng, B. Abeles, Y. Arie, Phys. Rev. Lett. 31 (1973) 44. C.J. Adkins, J. Phys. C 1 (1989) 1253. I.P. Zvyagin, R. Keiper, Phys. Stat. Sol. B 230 (2002) 151. A. Gerber, A. Milner, G. Deutscher, M. Karpovsky, A. Gladkikh, Phys. Rev. Lett. 78 (1997) 4277. T. Chui, G. Deutscher, P. Lindenfeld, W.L. McLean, Phys. Rev. B 23 (1981) 6172. I.S. Belobrodov, A.V. Lopatin, V.M. Vinokur, K.B. Efetov, Rev. Mod. Phys. 79 (2007) 469. K.B. Efetov, A. Tschersich, Phys. Rev. B 67 (2003) 174205. M. Gazda, Physica C 411 (2004) 170. M. Gazda, B. Kusz, S. Stizza, R. Natali, T. Klimczuk, L. Murawski, J. Non-Cryst. Sol. 353 (2007) 1023.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Electronic conduction in (Bi,Pb)–Sr–Ca–Cu–O granular superconductors M. Gazda *, B. Kusz, L. Murawski ´ sk, Poland Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdan
a r t i c l e
i n f o
Article history: Available online 31 July 2008 PACS: 74.72.Hs 72.20.Ee 74.81Bd Keyword: Electrical and electronic properties
a b s t r a c t Granular superconductors are very interesting materials owing to their untypical electrical properties caused by the presence of Coulomb effects, electron tunnelling, Josephson coupling between granules and vario\us aspects of disorder. The electrical properties of (Bi,Pb)–Sr–Ca–Cu–O granular superconductors obtained by the solid state crystallization method have been studied in this paper. The materials may be considered as a system of granules of a high-temperature suprconductor embedded in the insulating matrix. The main parameter which determines the properties of granular materials is the tunnelling conductivity between the neighboring grains. Composed of small weakly coupled granules they are characterized by hopping conductivity exponential dependence. Samples containing relatively large granules of the 2212 phase (above 30 nm), with larger inter-grain conductivity, above the transition temperature, exhibit a logarithmic temperature dependence of resistivity. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction
rðTÞ ¼ r0 1 a ln
Granular materials are composed of conducting grains of a size between a few and a few hundred nanometers embedded in an insulating or semiconducting matrix. They have been studied for many years and several physical mechanisms of electronic transport phenomena have been proposed [e.g. 1–6]. The main physical parameter which determines the properties of granular materials is tunnelling conductivity between neighboring grains [7]. It is described as dimensionless conductance, g, expressed in units of quantum conductance e2 = h. When the inter-grain conductance is low (g < 1), the granular material is in a dielectric regime and its electrical conductivity occurs through thermally activated tunnelling of electrons between grains [1]. Moreover, Coulomb effects may influence the electronic transport. The temperature dependence of the resistivity of granular materials in the dielectric regime usually follows the exponential relation:
qðTÞ ¼ q0 e
T0 T
n
;
ð1Þ
where n is in the range 0.2 6 n 6 1 and depends on the geometry of conductivity and the details of the density of states of the granular material. In low temperatures n at a level close to 0.5 has been often observed [3]. When conductivity between the grains is larger (g > 1), but still smaller than inter-grain conductivity, the granular material is in the metallic regime and Coulomb interaction between the grains may be neglected. In this case, as shown by Efetov et al., conductivity follows a logarithmic relation [8]: * Corresponding author. Tel.: +48 58 347 14 86; fax: +48 58 347 28 21. E-mail address: [email protected] (M. Gazda). 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.06.044
gEc ; T
ð2Þ
where Ec is the electrostatic energy of adding one electron to a grain,
a ¼ ð2pdgÞ1
ð3Þ
depends on the tunnelling conductivity of grains, g, and the system dimensionality, d. Dimensionality d is considered as half of the grains coordination number. The relations (2) and Eq. (3) are valid if T >> gd, where d is the mean energy level spacing in a single grain [8]. It should be also emphasized that the inter-grain conductivity depends on the granular material phase composition and microstructure. The behavior of a system in which grains transit into a superconducting state is not fully understood. Experimental studies have given a variety of results. For instance, a very interesting behavior of a ‘superconductor–insulator’ was found by Gerber et al. [5]. They observed that the resistivity of a granular metal below the critical temperature was much higher than in the normal state. Chui et al. reported three types of the temperature dependence of granular aluminium resistivity: exponential, /T1/2 and logarithmic [6]. A transition to a global superconducting state has been observed both in the case of strong localization with exponential temperature dependence of resistivity and in a weak localization regime. The presented work concerns the electrical properties of (Bi,Pb)4Sr3Ca3Cu4Ox granular materials obtained by solid state crystallization. These materials may be considered as a 3D system of grains of a metallic phase embedded in an insulating matrix. The
4324
M. Gazda et al. / Journal of Non-Crystalline Solids 354 (2008) 4323–4325
insulating or semiconducting matrix in which metallic grains are located is composed both of amorphous material and crystalline non-metallic phases [9]. The metallic phases which are formed as a result of crystallization are oxide superconductors belonging to the bismuth family, i.e. (Bi,Pb)2Sr2CuOx (2201 with Tc = 10 K), and (Bi,Pb)2Sr2CaCu2Ox (2212, Tc = 85 K). Depending on the temperature and time of crystallization, the size and/or the number of conducting grains increases while the width of the insulating barriers between them decreases. Simultaneously, normal state intergrain tunnelling conductivity increases and superconducting coupling between the grains becomes stronger. As a result, the electrical properties of the material change. 2. Experimental The samples were produced by solid state crystallization. Firstly, the samples of (Bi0.8Pb0.2)4Sr3Ca3Cu4Ox glass were prepared from a reagent grade: Bi(NO3) 5H2O, PbO, CuO, Sr(NO3)2 and CaCO3. The substrates were mixed at the (Bi,Pb):Sr:Ca:Cu 4:3:3:4 ratio and calcinated at 820 °C for 10 h. Next, they were melted in a platinum crucible at 1250 °C, kept at a high temperature for about 10 min and quenched. The glass was cut into bars of similar dimensions (2 1 8 mm3) and polished before further thermal treatment. The crystallization was carried out in a tube furnace at temperatures between 750 and 850 °C. The samples were put into a furnace which was already hot and after they were quenched after appropriate time. The heat treatment conditions affect the amount of metallic phases and the granule radius. Generally speaking, the higher the temperature and the longer the annealing time is, the greater is the number of metallic phases distributed in the form of larger granules in the sample [9]. The samples were checked by X-ray diffraction (XRD), atomic force microscopy (AFM) and scanning electron microscopy (SEM) methods. Measurements of resistivity as a function of temperature were made by a DC technique in a standard four-terminal configuration at temperatures ranging between 3 and 300 K. 3. Results and discussion Granular materials obtained by solid state crystallization may by divided into three groups. One group consists of samples in which metallic granules remain in a normal state in the whole analyzed temperature range. The second group includes materials in which granules transit into a superconducting state below the critical temperature, however the coupling between them is too weak and no long-range superconducting state develops. In materials belonging to the third group the grains are significantly larger than in other groups and the coupling between them is strong enough to make the whole sample superconduct below the critical temperature. It is not possible to calculate the exact value of tunnelling conductance g, or to determine the electrical properties of the granular system in a real granular system. Following the example shown by Efetov et al. [8] we have attempted to make some very rough estimations based on the knowledge of room temperature resistivity and average grain dimensions. The results are shown in Table 1. An analysis of the temperature dependence of resistivity in a (Bi,Pb)SrCaCuO granular system of small granules (not larger than 25 nm) being in a normal state has been performed elsewhere [10]. In this case the tunnelling conductivity between granules is small (g 1) and the temperature dependence of resistivity follows the exponential low (Fig. 1). It has been concluded that electronic transport occurs by the variable range hopping either with or without Coulomb interactions. Coulomb interactions determine con-
Table 1 Microstructural and electrical properties of the studied samples: d is the average diameter of (Bi,Pb)2Sr2CuOx or (Bi,Pb)2Sr2CaCu2Oy metallic granules; q(300 K) is the resistivity at the room temperature, g is the inter-grain dimensionless conductance and a is a parameter describing the logarithmic temperature dependence of resistivity Sample description
q(300 K)
Average grain dimension (nm)
Grain shape
d
g
a
(X cm)
500 °C, 500 °C, 650 °C, 750 °C, 800 °C, 820 °C, 840 °C,
1h 2h 4 min 2 min 2 min 2 min 2 min
10 ± 1 3.4 ± 0.3 1.6 ± 0.1 21 ± 0.2 6.6 ± 0.2 3.8 ± 0.2 0.25 ± 0.05
8–10 21–25 10–27 15–20 30–40 30–40 30–40
Oval Oval Oval Oval Oval Oval Oval
3 3 3 3 3 3 3
Exponential q(T) relation
850 °C, 2 min
0.45 ± 0.05
30–40
Oval
3
750 °C, 750 °C, 800 °C, 850 °C, 820 °C,
0.09 ± 0.01 0.06 ± 0.01 0.05 ± 0.01 0.03 ± 0.01 0.01 ± 0.001
100 100 1000 5000 5000
Oval Oval Plate Plate Plate
3 3 2 2 2
102 0.1 0.1 0.1 1 1 1– 10 1– 10 >10 >10 >100 >100 >100
8 min 32 min 8 min 32 min 16 min
0.156 0.158 0.156 0.156 0.134 0.134 0.128 0.128 0.1
Fig. 1. Resistivity of samples annealed at temperatures ranging between 500 and 750 °C plotted as a function of Tn.
ductivity in case of material containing granules of about 10 nm [10]. Examples of the temperature dependence of resistivity in a (Bi,Pb)SrCaCuO granular system of larger granules (equal to or larger than 30 nm) in the temperature ranging between 100 and 300 K are shown in Fig. 2. The condition T dg is fulfilled well, since d is approx. 0.02 meV for a grain dimension of about 20 nm and it decreases rapidly with dimensions of grains [10]. Also all the samples have significantly larger inter-grain conductivities than the samples with exponential q(T) dependence (Table 1). However, it can be seen that plots of resistivity, not conductivity as a function of ln T, are linear at quite a wide temperature range. The temperature dependence of resistivity in these samples may be expressed by the following relation:
qðTÞ ¼ Að1 a ln TÞ:
ð4Þ
Eqs. (4) and (2) are not identical, however at some temperature ranges they express similar dependencies. Therefore, it may be assumed that parameter a in (4) depends on the same properties of the granular system as the parameter in (2). Parameters describing the logarithmic temperature dependence of resistivity are collected in Table 1. It can be seen that parameter a tends to decrease with the increase of g, in qualitative agreement with the relation
M. Gazda et al. / Journal of Non-Crystalline Solids 354 (2008) 4323–4325
4325
4. Conclusions The studied materials were 3D systems of grains of (Bi,Pb)2Sr2CuOx and (Bi,Pb)2Sr2CaCu2Ox phases embedded in an insulating matrix. The electrical properties of (Bi,Pb)4Sr3Ca3Cu4Ox granular materials have been studied and discussed within the model of Efetov et al. The resistivity of the samples containing small, weakly coupled grains in a normal state follows the exponential dependence on temperature q(T) / exp[(T/T0)n]. Granular samples with larger inter-grain tunnelling conductivities exhibit a logarithmic temperature dependence of resistivity. Parameter a describing q(T) is in a qualitative agreement with the theoretical predictions. Fig. 2. Temperature dependence of resistivity of samples annealed at temperatures ranging between 750 and 850 °C.
Eq. (3). In order to quantitatively test the validity of relations (2), (4), a more precise way of determining the tunnelling conductance would be necessary. It is not clear why relation (4) is fulfilled better than relation (2) in a (Bi,Pb)SrCaCuO granular system. The reason may be connected with the inevitable distribution of grain dimensions and shapes. It should be also noted that such a logarithmic temperature dependence of resistivity has been recently observed in several granular materials [e.g. 5].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
B. Abeles, P. Sheng, M.D. Coutts, Y. Arie, Adv. Phys. 24 (1975) 407. P. Sheng, B. Abeles, Y. Arie, Phys. Rev. Lett. 31 (1973) 44. C.J. Adkins, J. Phys. C 1 (1989) 1253. I.P. Zvyagin, R. Keiper, Phys. Stat. Sol. B 230 (2002) 151. A. Gerber, A. Milner, G. Deutscher, M. Karpovsky, A. Gladkikh, Phys. Rev. Lett. 78 (1997) 4277. T. Chui, G. Deutscher, P. Lindenfeld, W.L. McLean, Phys. Rev. B 23 (1981) 6172. I.S. Belobrodov, A.V. Lopatin, V.M. Vinokur, K.B. Efetov, Rev. Mod. Phys. 79 (2007) 469. K.B. Efetov, A. Tschersich, Phys. Rev. B 67 (2003) 174205. M. Gazda, Physica C 411 (2004) 170. M. Gazda, B. Kusz, S. Stizza, R. Natali, T. Klimczuk, L. Murawski, J. Non-Cryst. Sol. 353 (2007) 1023.