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Thermal activation of carriers and characteristic features of the electronic structure of quasicrystalline systems
Materials Science and Engineering 294–296 (2000) 527–530
Thermal activation of carriers and characteristic features of the electronic structure of quasicrystalline systems A.F. Prekul∗ , N.Yu. Kuzmin, N.J. Shchegolikhina Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, S. Kovalevskaya Street, 18, Ekaterinburg, Russia Received 26 August 1999; accepted 7 January 2000
Abstract The electric conductivity and magnetic susceptibility of the quasicrystalline alloy Al62 Cu25.5 Fe12.5 are studied in the temperature range from 3.9 to 1200 K. The results are analysed under the assumption that the thermal activation of carriers is the main temperature effect. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystal; Electronic structure; Thermal activation
1. Introduction The physical properties of the stable i-phases are highly unusual and attract considerable interest of researchers. In recent years, a curious tendency is observed. The number of sophisticated models of electron structure and electron transport mechanisms increases. This activity is motivated by the statement that the transport properties of stable i-phases cannot be described as semiconducting (activated behavior) [1–3]. Briefly, the situation with temperature dependencies of electric conductivity σ (T) in this context is as follows. When presented in conventional coordinates ln σ (T) versus 1/T, the results of the direct experiment in the entire range of the i-phase existence (0–1200 K) do not form straight lines. Basov et al. [4] were probably the first, who succeeded in using the activation contributions for the σ (T) description in the quasicrystal Al70 Pd20 Re10 . There are two factors that promoted their success. This alloy has the lowest residual conductivity σ 0 or does not have it at all, in contrast to the other stable i-phases. On the other hand, according to the results of optical measurements the authors introduced into consideration three parameters of the carrier activation. Something similar is found in the quasicrystalline alloy Al62 Cu25.5 Fe12.5 after the subtraction of the residual conductivity from σ exp . The remaining temperature-dependent component σ t =σ exp −σ 0 , showed at once the tendency to the linearization in the coordinates ln σ (T) versus 1/T. In doing so, two linear sections are found with activation parameters which differ from one another by an order of magnitude [5]. ∗ Corresponding author. Tel.: +7-3432-499274; fax: +7-3432-745244. E-mail address: [email protected] (A.F. Prekul).
It is known that the electric conductivity is not the best property for the activation mechanism identification. The σ (T) behaviour depends not only on the carrier concentration n(T), but on their mobility µ(T) as well. If the latter does not follow the T−3/2 law, then the straightening coordinates in the general form ln(σ /Tα ) versus 1/T must be used. The pre-exponential factor Tα plays the greater role the smaller is the activation energy ∆. Obviously, the thermodynamic coefficients, which are independent of µ(T) are more convenient in this case. The study of the magnetic susceptibility χ (T) in the alloy Al62 Cu25.5 Fe12.5 [6] yielded a number of remarkable results. It is found that χ(T) and σ (T) curves are practically congruent. This clearly demonstrates that the carrier thermal activation is the main temperature effect. Furthermore, this clearly demonstrates that the thermally activated carriers form a non-degenerated electron gas. The corresponding analysis of the χ (T) curve showed that in the electron structure of the alloy there are either some gaps or one gap that increases rapidly with the temperature growth. The self-consistent consideration of χ(T) and σ (T) curves, the main results of which are presented here, showed that to the qualitative understanding and quantitative description of the experiment the presence of two energy gaps is sufficient.
2. Experimental An alloy with the nominal composition Al62 Cu25.5 Fe12.5 was prepared from high purity components in an arc furnace, quenched on the water-cooled heart by the ‘hammer and anvil’ method and annealed in a pure helium atmosphere
0921-5093/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 0 7 9 - 0
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A.F. Prekul et al. / Materials Science and Engineering 294–296 (2000) 527–530
at 730◦ C for 8 h. The X-ray diffraction analysis shows that the main phase after quenching is a β-solid solution with a CsCl lattice. An imperfect i-phase and possibly an amorphous phase are present in small quantities. The main phase after annealing is a rather perfect i-phase. In addition, there are weak indications of a cubic phase, as is typical of real, massive samples with this composition. The conductivity is measured by the standard four-probe method. Magnetic measurements were performed by two different devices. An MPMS-XL5 Quantum Design magnetometer was used in the temperature interval 3.9–400 K and a Domenicali balance was used in the interval 300–1100 K. The overlapping of the measurement ranges was used to match the low and high temperature results.
component of the magnetic susceptibility has the form ∆ ; exp − T
χi = AT
1/2
3/2 1/2
A = 2me kB (2π ~2 )3/2 µ2B (1)
(all the notations are standard) 3.1. χ (T) analysis
3. Results
Actually, we took into account the availability of two activation parameters ∆1 and ∆2 and used the χ(T) dependence in the form ∆1 ∆2 1/2 1/2 + A2 T exp − (2) χi = χ0 + A1 T exp − T T
The results are shown in Fig. 1. Here, curve 1 is the magnetic susceptibility (χ i ). It is obtained by subtracting a Curie–Weiss type contribution (χ C−W ) associated either with remains of the cubic phase and the magnetic moment of Fe-atoms in this phase or with uncontrolled impurities. Curve 2 shows the electric conductivity. As one can see, the variation of the magnetic susceptibility as a function of temperature is similar to the variation of the electric conductivity. When being superposed, the curves virtually coincide. Therefore, we can safely speak about the congruence of the curves χ (T) and σ (T). This correlation is a strong argument for the fact that the change in the concentration of charge-carriers is the main temperature effect determining the behaviour of the transport and thermodynamic coefficients. If this is so, it becomes interesting that the paramagnetism of the system becomes stronger as χ changes with increasing temperature. This is easy to understand assuming that the thermally activated carriers form a non-degenerated electron gas. Then, as is well known [7], the temperature-dependent
The main difficulty lies in the finding of the separation technique of different contributions. The preliminary modelling shows that under the conditions (∆2 /∆1 )≤10, the influence of the χ1∆ in the high-temperature region might not be neglected. On the contrary, in the intermediate and low-temperature regions the influence of the χ2∆ contribution may be neglected. On the other hand, the experimental curves χ i (T) and σ (T) have a visually variable curvature. At intermediate temperatures the curves are more flat than at lower and higher temperatures. The modelling shows that the approximation of the flat portion of the curve gives the most suitable criteria for determining ∆1 in the coordinates ln[(χi − χ0∗ )/T 1/2 ] versus 1/T, (χ0∗ is the fitting parameter). After determining ∆1 the contribution χ2∆ = χi − χ0∗ − χ1∆ is calculated and the ∆2 is determined from the dependence ln[χ2∆ /T 1/2 ] versus 1/T. Within experimental error, the activation energies are as follows: ∆1 =460±20 K, ∆2 =2700±50 K. In Fig. 2 we present all the contributions, i.e. χ C−W , χ1∆ and χ2∆ , and the description of the whole experiment.
Fig. 1. Magnetic susceptibility (1) and electric conductivity (2) dependencies of the icosahedral phase Al62 Cu25.5 Fe12.5 are practically congruent.
Fig. 2. Magnetic susceptibility of the real alloy Al62 Cu25.5 Fe12.5 (䊊) and its description (solid line) by three components: C−W (1), χ1∆ (2) and χ2∆ (3).
A.F. Prekul et al. / Materials Science and Engineering 294–296 (2000) 527–530
Fig. 3. Electric conductivity of the real alloy Al62 Cu25.5 Fe12.5 (䊊) and its description (solid line) by three components: σ ML =σ 0 +BT1/2 (1), σ1∆ (2), and σ2∆ (3).
3.2. σ (T) analysis The purpose of the analysis was to check with such determined parameters ∆1 and ∆2 the conductivity σ (T) and to determine the value of α. We proceed from the fact that the real σ (T) dependence in most quasicrystals may be represented in the form of the dependence involving two components, namely, metal-like (ML) and semiconductor-like (SCL) components, i.e. σ (T)=σ ML +σ SCL . The ML component usually contains the ‘root-square’ correction and is of the form σ ML =σ 0 +BT1/2 (σ 0 is the residual conductivity of a system). In its turn, 460 2700 + A2 T α2 exp − (3) σSCL = A1 T α1 exp − T T The fitting of these formula by parts is similar to the above described technique. The results of the fitting, the quality of which could be judged by Fig. 3, were found to be as follows: α 1 =0.4, α 2 =1.6. This means that the mobility of carriers with the activation energy ∆1 follows the law µ1 ∼T−1 . The mobility of carriers with the activation energy ∆2 follows the law µ2 ∼T0 , and therefore, is temperature independent. Note that the numerical evaluations of α 1 and α 2 are obtained in the assumption that σ ML =constant. This simplifies the analysis, but causes the calculated curve to pass slightly above the experimental curve at T<200 K. A consideration of σ ML in the form σ 0 +BT1/2 removed this discrepancy while leaving the values of α 1 and α 2 unchanged. (The positive sign of the coefficient B in this ‘root-square’ correction will be discussed elsewhere).
529
thermally activated carriers. The ratio σ /χ gives the mobility. The results are as follows: µ1 (300)=60 cm2 v−1 s−1 , µ2 =17 cm2 v−1 s−1 . The ratios A1 /A and A2 /A give the carrier effective masses, m∗1 = 2.1me , m∗2 = 10.4me . Correspondingly, the typical relaxation times of carriers are τ 1 (300)=7×10−14 s, τ 2 =9×10−14 s. According to these estimations there are neither anomalously strong scattering nor anomalously short relaxation times of thermally activated charge carriers in the quasicrystalline alloy under consideration. Thus, from our analysis it may be deduced that there are two groups of carriers separated from the electron state continuum by the energy barriers 0.04 eV and 0.25 eV in values. The nature of these barriers is not yet understood. Since the relationships of the classical semiconductor physics were used to reveal these barriers, it is possible to assume that they are the ordinary band gaps associated with the lattice symmetry. Therefore, this feature of the electron structure may be an inherent characteristic of all the stable i-phases containing transition d-metals. In support of this conjecture it is interesting to compare a situation in i-phase Al70 Pd20 Re10 [4]. Two activation barriers 0.03 eV and 0.26 eV were determined in it from optical experiments. In this sense, it is a good agreement. As for the methods capable to reach a good description of σ (T) in a wide range of temperatures 10–1000 K, they are different. In [4] such a description is reached with addition of the variable range hopping term at T<200 K. We made it through the preexponential factor Tα . We believe that the formula (3) is rather universal and available for description rather complex σ (T) dependencies. As an example, Fig. 4 shows the types of dependence of the electric conductivity that may be expected for different value ∆1 , all other factors being the same. In particular, in the limit of very small ∆1 at low temperatures, the electric conductivity may follow to the law close to T1/2 in the absence of quantum corrections to σ ML .
4. Discussion We were able to separate thermal activation contributions both in σ and in χ. Subsequently, it is easy to obtain numerical values of the main characteristics of the
Fig. 4. Types of σ (T) curves for different ratios: (a) ∆2 /∆1 :=2700/200; (b) ∆2 /∆1 =2700/460; (c) ∆2 /∆1 =2700/800.
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A.F. Prekul et al. / Materials Science and Engineering 294–296 (2000) 527–530
5. Conclusion We measured temperature dependencies of electric conductivity and magnetic susceptibility of the quasicrystalline alloy Al62 Cu25.5 Fe12.5 and analysed them under the assumption of the decisive role of the thermal carrier activation mechanism in the physical properties of this alloy. By and large, we showed that the relationships of the classical semiconductor physics are well suited for the qualitative understanding and for the quantitative description of temperature dependencies of the kinetic and thermodynamic coefficients of the quasicrystalline systems.
References [1] D. Mayou, C. Berger, F. Cyrot-Lackman, T. Klein, P. Lanco, Phys. Rev. Lett. 70 (1993) 3915. [2] C. Janot, Phys. Rev. B 53 (1996) 181. [3] R. Haberkern, G. Fritsch, in: C. Janot, R. Mosseri (Eds.), Proceedings of the Fifth International Conference on Quasicrystals, Avignon, 22–26 May 1995, World Scientific, Singapore, p. 460. [4] D.N. Basov, F.S. Pierce, P. Volkov, S.J. Poon, T. Timusk, Phys. Rev. Lett. 73 (1994) 1865. [5] A.F. Prekul, L.V. Nomerovannaya, A.B. Rol’shchikov, N.I. Shchegolikhina, S.V. Yartsev, Phys. Met. Metallogr. 82 (1996) 479. [6] A.F. Prekul, N.Y. Kuzmin, N.I. Shchegolikhina, JETP Letters 69 (1999) 221. [7] S.V. Vonsovskii, Magnetism, FB, Nauka, Moscow, 1971, p. 238, (in Russian).
Thermal activation of carriers and characteristic features of the electronic structure of quasicrystalline systems A.F. Prekul∗ , N.Yu. Kuzmin, N.J. Shchegolikhina Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, S. Kovalevskaya Street, 18, Ekaterinburg, Russia Received 26 August 1999; accepted 7 January 2000
Abstract The electric conductivity and magnetic susceptibility of the quasicrystalline alloy Al62 Cu25.5 Fe12.5 are studied in the temperature range from 3.9 to 1200 K. The results are analysed under the assumption that the thermal activation of carriers is the main temperature effect. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystal; Electronic structure; Thermal activation
1. Introduction The physical properties of the stable i-phases are highly unusual and attract considerable interest of researchers. In recent years, a curious tendency is observed. The number of sophisticated models of electron structure and electron transport mechanisms increases. This activity is motivated by the statement that the transport properties of stable i-phases cannot be described as semiconducting (activated behavior) [1–3]. Briefly, the situation with temperature dependencies of electric conductivity σ (T) in this context is as follows. When presented in conventional coordinates ln σ (T) versus 1/T, the results of the direct experiment in the entire range of the i-phase existence (0–1200 K) do not form straight lines. Basov et al. [4] were probably the first, who succeeded in using the activation contributions for the σ (T) description in the quasicrystal Al70 Pd20 Re10 . There are two factors that promoted their success. This alloy has the lowest residual conductivity σ 0 or does not have it at all, in contrast to the other stable i-phases. On the other hand, according to the results of optical measurements the authors introduced into consideration three parameters of the carrier activation. Something similar is found in the quasicrystalline alloy Al62 Cu25.5 Fe12.5 after the subtraction of the residual conductivity from σ exp . The remaining temperature-dependent component σ t =σ exp −σ 0 , showed at once the tendency to the linearization in the coordinates ln σ (T) versus 1/T. In doing so, two linear sections are found with activation parameters which differ from one another by an order of magnitude [5]. ∗ Corresponding author. Tel.: +7-3432-499274; fax: +7-3432-745244. E-mail address: [email protected] (A.F. Prekul).
It is known that the electric conductivity is not the best property for the activation mechanism identification. The σ (T) behaviour depends not only on the carrier concentration n(T), but on their mobility µ(T) as well. If the latter does not follow the T−3/2 law, then the straightening coordinates in the general form ln(σ /Tα ) versus 1/T must be used. The pre-exponential factor Tα plays the greater role the smaller is the activation energy ∆. Obviously, the thermodynamic coefficients, which are independent of µ(T) are more convenient in this case. The study of the magnetic susceptibility χ (T) in the alloy Al62 Cu25.5 Fe12.5 [6] yielded a number of remarkable results. It is found that χ(T) and σ (T) curves are practically congruent. This clearly demonstrates that the carrier thermal activation is the main temperature effect. Furthermore, this clearly demonstrates that the thermally activated carriers form a non-degenerated electron gas. The corresponding analysis of the χ (T) curve showed that in the electron structure of the alloy there are either some gaps or one gap that increases rapidly with the temperature growth. The self-consistent consideration of χ(T) and σ (T) curves, the main results of which are presented here, showed that to the qualitative understanding and quantitative description of the experiment the presence of two energy gaps is sufficient.
2. Experimental An alloy with the nominal composition Al62 Cu25.5 Fe12.5 was prepared from high purity components in an arc furnace, quenched on the water-cooled heart by the ‘hammer and anvil’ method and annealed in a pure helium atmosphere
0921-5093/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 0 7 9 - 0
528
A.F. Prekul et al. / Materials Science and Engineering 294–296 (2000) 527–530
at 730◦ C for 8 h. The X-ray diffraction analysis shows that the main phase after quenching is a β-solid solution with a CsCl lattice. An imperfect i-phase and possibly an amorphous phase are present in small quantities. The main phase after annealing is a rather perfect i-phase. In addition, there are weak indications of a cubic phase, as is typical of real, massive samples with this composition. The conductivity is measured by the standard four-probe method. Magnetic measurements were performed by two different devices. An MPMS-XL5 Quantum Design magnetometer was used in the temperature interval 3.9–400 K and a Domenicali balance was used in the interval 300–1100 K. The overlapping of the measurement ranges was used to match the low and high temperature results.
component of the magnetic susceptibility has the form ∆ ; exp − T
χi = AT
1/2
3/2 1/2
A = 2me kB (2π ~2 )3/2 µ2B (1)
(all the notations are standard) 3.1. χ (T) analysis
3. Results
Actually, we took into account the availability of two activation parameters ∆1 and ∆2 and used the χ(T) dependence in the form ∆1 ∆2 1/2 1/2 + A2 T exp − (2) χi = χ0 + A1 T exp − T T
The results are shown in Fig. 1. Here, curve 1 is the magnetic susceptibility (χ i ). It is obtained by subtracting a Curie–Weiss type contribution (χ C−W ) associated either with remains of the cubic phase and the magnetic moment of Fe-atoms in this phase or with uncontrolled impurities. Curve 2 shows the electric conductivity. As one can see, the variation of the magnetic susceptibility as a function of temperature is similar to the variation of the electric conductivity. When being superposed, the curves virtually coincide. Therefore, we can safely speak about the congruence of the curves χ (T) and σ (T). This correlation is a strong argument for the fact that the change in the concentration of charge-carriers is the main temperature effect determining the behaviour of the transport and thermodynamic coefficients. If this is so, it becomes interesting that the paramagnetism of the system becomes stronger as χ changes with increasing temperature. This is easy to understand assuming that the thermally activated carriers form a non-degenerated electron gas. Then, as is well known [7], the temperature-dependent
The main difficulty lies in the finding of the separation technique of different contributions. The preliminary modelling shows that under the conditions (∆2 /∆1 )≤10, the influence of the χ1∆ in the high-temperature region might not be neglected. On the contrary, in the intermediate and low-temperature regions the influence of the χ2∆ contribution may be neglected. On the other hand, the experimental curves χ i (T) and σ (T) have a visually variable curvature. At intermediate temperatures the curves are more flat than at lower and higher temperatures. The modelling shows that the approximation of the flat portion of the curve gives the most suitable criteria for determining ∆1 in the coordinates ln[(χi − χ0∗ )/T 1/2 ] versus 1/T, (χ0∗ is the fitting parameter). After determining ∆1 the contribution χ2∆ = χi − χ0∗ − χ1∆ is calculated and the ∆2 is determined from the dependence ln[χ2∆ /T 1/2 ] versus 1/T. Within experimental error, the activation energies are as follows: ∆1 =460±20 K, ∆2 =2700±50 K. In Fig. 2 we present all the contributions, i.e. χ C−W , χ1∆ and χ2∆ , and the description of the whole experiment.
Fig. 1. Magnetic susceptibility (1) and electric conductivity (2) dependencies of the icosahedral phase Al62 Cu25.5 Fe12.5 are practically congruent.
Fig. 2. Magnetic susceptibility of the real alloy Al62 Cu25.5 Fe12.5 (䊊) and its description (solid line) by three components: C−W (1), χ1∆ (2) and χ2∆ (3).
A.F. Prekul et al. / Materials Science and Engineering 294–296 (2000) 527–530
Fig. 3. Electric conductivity of the real alloy Al62 Cu25.5 Fe12.5 (䊊) and its description (solid line) by three components: σ ML =σ 0 +BT1/2 (1), σ1∆ (2), and σ2∆ (3).
3.2. σ (T) analysis The purpose of the analysis was to check with such determined parameters ∆1 and ∆2 the conductivity σ (T) and to determine the value of α. We proceed from the fact that the real σ (T) dependence in most quasicrystals may be represented in the form of the dependence involving two components, namely, metal-like (ML) and semiconductor-like (SCL) components, i.e. σ (T)=σ ML +σ SCL . The ML component usually contains the ‘root-square’ correction and is of the form σ ML =σ 0 +BT1/2 (σ 0 is the residual conductivity of a system). In its turn, 460 2700 + A2 T α2 exp − (3) σSCL = A1 T α1 exp − T T The fitting of these formula by parts is similar to the above described technique. The results of the fitting, the quality of which could be judged by Fig. 3, were found to be as follows: α 1 =0.4, α 2 =1.6. This means that the mobility of carriers with the activation energy ∆1 follows the law µ1 ∼T−1 . The mobility of carriers with the activation energy ∆2 follows the law µ2 ∼T0 , and therefore, is temperature independent. Note that the numerical evaluations of α 1 and α 2 are obtained in the assumption that σ ML =constant. This simplifies the analysis, but causes the calculated curve to pass slightly above the experimental curve at T<200 K. A consideration of σ ML in the form σ 0 +BT1/2 removed this discrepancy while leaving the values of α 1 and α 2 unchanged. (The positive sign of the coefficient B in this ‘root-square’ correction will be discussed elsewhere).
529
thermally activated carriers. The ratio σ /χ gives the mobility. The results are as follows: µ1 (300)=60 cm2 v−1 s−1 , µ2 =17 cm2 v−1 s−1 . The ratios A1 /A and A2 /A give the carrier effective masses, m∗1 = 2.1me , m∗2 = 10.4me . Correspondingly, the typical relaxation times of carriers are τ 1 (300)=7×10−14 s, τ 2 =9×10−14 s. According to these estimations there are neither anomalously strong scattering nor anomalously short relaxation times of thermally activated charge carriers in the quasicrystalline alloy under consideration. Thus, from our analysis it may be deduced that there are two groups of carriers separated from the electron state continuum by the energy barriers 0.04 eV and 0.25 eV in values. The nature of these barriers is not yet understood. Since the relationships of the classical semiconductor physics were used to reveal these barriers, it is possible to assume that they are the ordinary band gaps associated with the lattice symmetry. Therefore, this feature of the electron structure may be an inherent characteristic of all the stable i-phases containing transition d-metals. In support of this conjecture it is interesting to compare a situation in i-phase Al70 Pd20 Re10 [4]. Two activation barriers 0.03 eV and 0.26 eV were determined in it from optical experiments. In this sense, it is a good agreement. As for the methods capable to reach a good description of σ (T) in a wide range of temperatures 10–1000 K, they are different. In [4] such a description is reached with addition of the variable range hopping term at T<200 K. We made it through the preexponential factor Tα . We believe that the formula (3) is rather universal and available for description rather complex σ (T) dependencies. As an example, Fig. 4 shows the types of dependence of the electric conductivity that may be expected for different value ∆1 , all other factors being the same. In particular, in the limit of very small ∆1 at low temperatures, the electric conductivity may follow to the law close to T1/2 in the absence of quantum corrections to σ ML .
4. Discussion We were able to separate thermal activation contributions both in σ and in χ. Subsequently, it is easy to obtain numerical values of the main characteristics of the
Fig. 4. Types of σ (T) curves for different ratios: (a) ∆2 /∆1 :=2700/200; (b) ∆2 /∆1 =2700/460; (c) ∆2 /∆1 =2700/800.
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A.F. Prekul et al. / Materials Science and Engineering 294–296 (2000) 527–530
5. Conclusion We measured temperature dependencies of electric conductivity and magnetic susceptibility of the quasicrystalline alloy Al62 Cu25.5 Fe12.5 and analysed them under the assumption of the decisive role of the thermal carrier activation mechanism in the physical properties of this alloy. By and large, we showed that the relationships of the classical semiconductor physics are well suited for the qualitative understanding and for the quantitative description of temperature dependencies of the kinetic and thermodynamic coefficients of the quasicrystalline systems.
References [1] D. Mayou, C. Berger, F. Cyrot-Lackman, T. Klein, P. Lanco, Phys. Rev. Lett. 70 (1993) 3915. [2] C. Janot, Phys. Rev. B 53 (1996) 181. [3] R. Haberkern, G. Fritsch, in: C. Janot, R. Mosseri (Eds.), Proceedings of the Fifth International Conference on Quasicrystals, Avignon, 22–26 May 1995, World Scientific, Singapore, p. 460. [4] D.N. Basov, F.S. Pierce, P. Volkov, S.J. Poon, T. Timusk, Phys. Rev. Lett. 73 (1994) 1865. [5] A.F. Prekul, L.V. Nomerovannaya, A.B. Rol’shchikov, N.I. Shchegolikhina, S.V. Yartsev, Phys. Met. Metallogr. 82 (1996) 479. [6] A.F. Prekul, N.Y. Kuzmin, N.I. Shchegolikhina, JETP Letters 69 (1999) 221. [7] S.V. Vonsovskii, Magnetism, FB, Nauka, Moscow, 1971, p. 238, (in Russian).